# Extended Dimensionality of Interval cmavo

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Presently, the cmavo "mi'i", "bi'i", "bi'o" (which currently constitute all elements of selma'o BIhI) all represent/create one-dimensional intervals. However, in mathematics and even in daily life, there are many instances when higher-dimensional intervals are desired. This functionality should be supported.

Rather than creating new cmavo for this task, the current cmavo (aforementioned) can simply be extended. The proposal described here will have the objective of supporting functionality for description of higher-dimensional intervals via extension only; only mathematical points are being discussed. The result should be back-compatible.

## Contributors

The following people have contributed to the writing of this article (or have provided ideas for it):

- lai .krtisfranks.

Where first-person personal pronouns are used, they refer to at least one of these individuals. However, they will be avoided whensoever possible (with explicit mention of the author's name when a personal opinion or insight is conveyed).

## Current Functionality

The cmavo of BIhI are nonlogical interval connectives. In mathematics (other options are available), one inputs a real number or possibly a generic endpoint, follows it by a cmavo of BIhI, and then mentions another real number or endpoint. The result is a description of a set of all points belonging to an interval. More explicitly:

Let x and y be real numbers or points in geometric space, let r be a nonnegative real number. Let the space to which x and y belong be X. Further suppose that X does not "loop around" in any sense. Then:

- x mi'i r: generates the interval centered on x which has range r on either side of x; in other words, this is the set of all points that have a distance to x that is less than (or possibly equal to) r. x is the midpoint of the interval, which has length 2r. Such a thing is sometimes denoted in a fashion similar to <math> \operatorname{B}_1 (x, r) </math>, where "B" is for "ball" and the subscript "1" tells the dimensionality of the space; this is also called an r neighborhood of x (sometimes denoted <math> \operatorname{nbhd} (x, r) </math>), where the space is inferred from context.
- x bi'i y: generates the interval or unordered line segment with endpoints x and y; in other words, this is the set of all points between x and y (possibly including either, both, or neither of the endpoints). "y bi'i x" is completely equivalent to "x bi'i y"; there is no inherent order to the inputs nor direction to the line segment. This is the generic meaning for "between" when referring to an interval, in normal life or in mathematics; there is no notation for this notion which is both commonly understood and known to lai .krtisfranks. ; supposing an ordering on X, the closest thing would be an interval of form: <math>(\operatorname{min}(\{x,y\}), \operatorname{max}(\{x,y\})) \cup A</math>, where <math>A \subseteq \{x,y\}</math>. However, if X is a partially ordered space with order relation '<', then we may describe it thusly: let 'R' denote either '<' or its complement/negative '>'; then, if the endpoints are excluded, "x bi'i y" generates the set <math> \{ \alpha \in X: x </math> R <math> \alpha </math> R <math> y \} </math>. (Notice here that x and y may be presented in either order but for any given selection of presentation order, 'R' is fixed in meaning and present in both relations; if one order of presentation is true, then if the order is switched, then the resulting statement will mean the same thing but the meaning of 'R' will be changed to the other inequality relation. If an endpoint is to be included, this set will just be united with the singleton set of that endpoint. If X cannot be or is not partially ordered, then this present discussion about mathematical representation may be ignored; in such cases, this BIhI construction may still make sense, however - just revert to a more intuitive understanding based on the English description).
- x bi'o y: generates the interval or 'directed' line segment with endpoints x and y in that order (starting from x and going to y); otherwise, it is equivalent to "bi'i". This is the typical meaning of intervals of form [x, y) and the like. Thus "y bi'o x" is backward relative to "x bi'o y". Continuing the discussion in the immediately previous point about "bi'i" which was concerned with nathematical representation of the construct formed, and supposing the same conditions and notation, then "x bi'o y" is exactly the same (and the same commentary applies) except that it demands that 'R' represents '<'. (Notice additionally that, in this case, x < y must be true. However be careful to avoid reading "<" as the symbol representing "less than" in the intuitive sense for real numbers; it could just as easily be any other partial order, including the "greater than" relation).

When X does not "loop around" through the initially-potential intervals in question (it may do so through others): "bi'i" has a symmetry between its two arguments. Thus, it would be weird for conditions to be placed on exactly one of its arguments. "bi'o" is a restriction of "bi'i" which forces a directionality or order upon the line segment produced; thus "bi'o" inherits properties from "bi'i" but the broken symmetry allows conditions to be placed on any combination of its arguments in a natural context.

If X does "loop around" through at least one initially-potential interval at hand, then: "bi'i" forces the 'most natural' or 'minimal' interval to be the one which is being referenced under either ordering of the inputes, and "bi'o" results in the 'most natural' or 'minimal' interval which extends **from** the first argument **to** the second argument. For example, on a circle, where angles are measured counterclockwise (from the positive x-axis, toward the positive y-axis; an angle of measure <math>0</math> has its rays both being equivalent to the positive x-axis) and where the arguments refer to points on the circle by the angles so measured from the positive x-axis (in radians) at which they are located (modulo <math>\tau = C/r</math> for circle of circumference <math>C</math> and radius <math>r>0</math>): "<math>0</math> bi'o <math>\tau/4</math>" is equivalent to "<math>0</math> bi'i <math>\tau/4</math>" (and, therefore, "<math>\tau/4</math> bi'i <math>0</math>", which is the single arc segment which forms one quarter of the entire circle and which is between the positive x-axis and the positive y-axis; but "<math>\tau/4</math> bi'o <math>0</math>" is entirely different, being the other three quarters of the circle (and, if it matters, this is traced out via having the angle run counterclockwise from positive y-axis (at <math>\tau/4</math>) until it attains the value <math>\tau \equiv 0</math> at the positive x-axis (from 'the other side', so to speak). Throughout this article, X will typically be assumed to not 'loop around' through any initially-potential interval at hand, but the considerations made in this paragraph should apply without too much difficulty in making the analogy.

- (Note that "initially-potential interval" is interpreted broadly, accounting for all conceivable references intended for the interval at hand, which means that X 'loop around' through at least one of the results of "bi'o" and of "se bi'o" for the given arguments (in fixed order); the set of such potential intervals are those which could be meant prior to the audience actually thinking too deeply about the meaning/structure (it is the set of all initial interpretations which are conceivable, regardless of whether they are possible). In this sense, an interval is a path from one of the arguments to the other. Consider a graph/network G of points with some edges. It might be the case that G is a tree (lacks any loops) except for, say, exactly one loop (a sequence of edge-connected nodes such that there is at least one path from at least one node to itself via the edges) which includes a proper subset of its nodes, numbering at least two (for simplicity). As long as both "bi'o" and "se bi'o" do not intersect nonemptily with any of the nodes/edges in this loop, then the loop can be 'excised' from consideration and we can take X to not loop around through any of the initially-potential intervals; in other words X would be the result of 'subtracting' the loop from G; such a subtraction would have no impact on the interval actually meant. On the other hand, if at least one of those intervals does indeed intersect nonemptily with the loop, then X must be taken to 'loop around' through at least one initially-potential interval at hand, and this potential 'looping' must be taken into account when interpreting/evaluating the meaning of the interval being specified).

Suppose that X does not loop through any initially-potential interval at hand.

It should be noted that, unless <math>x = \pm \infty</math> is defined and true in a given case, <math>\forall r \in [0, + \infty]</math>, "x mi'i r" refers to/forms a bounded interval or the whole space; no semi-infinite (that is: bounded from above/the positive side xor from below/the negative side) interval/line segment (ray) can be produced. The same is true for "bi'i" and "bi'o" if, additionally, r is finite; recall the previous commentary about symmetry of arguments and inheritance concerning these words. These proposals - where '<math>\infty</math>' now refers to any point which has a distance of <math>+ \infty</math> from the origin - will (or, at least, thus far have) not change(d) this fact. In both the current (non-proposed) functionality and the herein proposed functionality, the intervals which are produced by "bi'i" and "bi'o" can be bounded, semi-infinite (in any number of directions (so long as that number is a nonnegative integer less than or equal to the dimensionality of the space)), or infinite in both positive and negative directions (doubly-infinite/infinite; in any number of directions (so long as that number is a nonnegative integer less than or equal to twice the dimensionality of the space)); in the herein proposed functionality, these states can be simultaneously true, with the interval being semi-infinite in some dimensions, infinite in others, and bounded/finite in yet others (the total count of such dimensions being equal to the dimensionality of the space).

If we accept partial orders, the space X can be all sorts of creatures, including - for example - sets under the strict-containment/proper-subset relation (so that BIhI forms an interval of sets). However, more exotic meanings can be used/intended (although any partial order endowing the space would have to be ignored in context with respect to the meaning of BIhI, which is okay and implicitly possible within the description heretofore provided by the CLL). For example, intervals may just trace out (a possibly ordered/'directed') path between points in X, which may be - for example - the geography of locations on Earth, a network, or a set of sets (which may otherwise but inconsequentially for our purposes be endowed with the proper-subset order). In order to be clear: X need not have an order of any kind endowing it overall; however, if "bi'o" is used, the interval generated does have an ordered endowed on it (alone) which may or may not match the order endowing X, should such an order exist.

A final note for the sake of carefulness: "bi'o" establishes an order on a line segment such that its first argument is somehow 'less' than its second one. However, it does not necessarily/really establish a direction on the line segment in a graph theoretic sense. So, throughout this page, take mentions of directionality with a grain of salt - they may be the result of momentary carelessness.

### SE BIhI

Although independent of this whole proposal, SE (specifically "se" and "re'au'e") should be able to precede BIhI. For "bi'i" itself, there would be no effect. For "se bi'o", the order of the arguments is switched so that the first argument of "se bi'o" is the greater/destination endpoint and the second argument thereof is the lesser/origin endpoint. Thus, "bi'i" is equivalent to "bi'o ja se bi'o". For "se mi'i", the order of the arguments is switched so that the first represents the radius length (in one-dimension: half of the length of the linear interval) and the second represents the center of the interval.

## Proposed Functionality

Any commentary in this article (excluding the 'Authors' and 'Current Functionality' sections) is meant to be taken as part of the "whole proposal". What follows are specific details which are broken into labelled sections for the sake of reference and hierarchy of application.

### Main Proposal #1

Fix a space X which is endowed with a metric d and defined over an ordered field F which is also endowed with a compatible metric <math>d_F</math>; fix a basis B thereof. Let r be a nonnegative real number. Let x and y live in the same space X. Define the dimensionality of X to be dim(X) = n, where n is any nonnegative integer or (for simplicity: countable) infinity. Define B = {<math>e_1, e_2, ..., e_n</math>}. Then there exists <math> x_1, x_2, ..., x_n, y_1, y_2, ..., y_n \in F: x = x_1 e_1 +...+ x_n e_n = (x_1, ..., x_n), y = y_1 e_1 +...+ y_n e_n = (y_1, ..., y_n)</math>. Let "*R*" denote an 'ordering' relation on F or the ordered field of real numbers (as appropriate) which may be either the "less than" relation (denoted "<") or, as appropriate (determined by GAhO; generically, elliptical), the "less than or equal to" relation (denoted "<math>\leq</math>").

We let the dimensionality of our space (which is and can be inferred from the dimensionality of x and/or y, which should match) determine the nature of our intervals.

- "mi'i" is the easiest to redefine. In fact, the previous description needs no reworking, so long as we understand the space to be potentially larger than a line and loosen our notion of "interval". I propose that "x mi'i r" is defined to be/describe the n-dimensional hyperball (or, possibly, the closure thereof) which is centered on/at x and which has radius r. Notationally, it is <math>\{ z \in X: d(x,z)</math>
*R*<math>r \}</math>.- In order to be clear: this interval is a neighborhood in the space; that is to say, it is a region of the space which is bounded by and internal to a sphere (but which may possibly include the boundary and/or exclude the center). This region is called a "ball".
- I propose that we adopt additional keywords/glosses/terminology for "mi'i". "mi'i" should be given the keyword/gloss "centered interval"; it might also deserve the keyword/gloss "n-ball". The second argument (here denoted by "r") should be called the "radius" (in addition to "range"). The first argument (here denoted by "x") can remain with the sole label of "center".

- "bi'i" requires a little more work. I propose that "x bi'i y" generates/describes the n-cell/n-orthotope which has opposite vertices at points x and y. This is <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{N} \cap [1, n]), (x_i</math>
*R*<math> \alpha_i </math>*R*<math> y_i)) \}</math>.- This is similar to the rectangle made on a computer by clicking the mouse at one endpoint and holding-with-dragging the cursor to the other endpoint. Note that there are as many ways to generate the same 'rectangle' as there are vertices on/of the 'rectangle' (where this jumber scales with dimensionality of the 'recrangle'). It need not be two-dimensional, though.
- Terminology can again be updated (id est: added to). The interval should be additionally described as a "n-cell" and "n-orthotope interval"; "rectilinear interval" may additionally be considered. Both arguments (here denoted by "x" and "y" respectively) should be labelled as "endpoints". Symmetry between them should be noted in dictionary definitions.
- This extended form of "bi'i" can be obtained via Cartesian products of linear intervals aligned with/generated by the basis elements. We will exploit this fact in the discussion about the endpoint stati (see the section named accordingly).

- "bi'o" has, to me (lai krtisfranks), no obvious extension since (for example) <math>\mathbb{R}^2</math> cannot be ordered.

_

When <math>F = \mathbb{R}</math>, 1-tuples/1-dimensional endpoints will be isomorphically mapped automatically to the corresponding real numbers. This allows for ease of use and back-compatibility.

- This is done by establishing the identity/correspondence that <math>\forall (\xi) \in F, (\xi) \leftrightarrow \xi </math>. In other words, the functionality of intervals as defined outside of this whole proposal (which is for the 1-dimensional case) is extended so that endpoints "x" and "y" which are scalars are automatically mapped to (x) and (y) respectively, where the latter are now handled via the extended functionality herein proposed (as a point in (albeit one-dimensional) space and which has coordinates (well, exactly 1 coördinate)).

Notice that, now, "mi'i" and "bi'i" do not have the same "shape" except when the dimensionality involved is 0 or 1. The former is round whereas the latter is rectilinear. This is assuming Euclidean geometry. If other netrics are involved, they may appear to be the same or may actually be the same. For example, in taxicab geometry, a sphere *appears* to be a cross-polytope of the appropriate dimensionality, yet it is still a sphere (which bounds "mi'i"-intervals). In Chebyshev geometry, a sphere appears as an orthotope of the appropriate dimensionality, yet it is still a spehere (and the boundary of a "mi'i"-interval); in this case, though, it very well may be congruent to the n-cell (orthotope) that "bi'i" produces (under the proper conditions).

- These distinct definitions are good (utile) and natural in theoretical mathematics.

### Alternative #1: Line Segments Unless Specified Otherwise

An alternative (which lai .krtisfranks. finds perhaps even better than the previous proposal (Main Proposal #1)) is to have "bi'i" and "bi'o" always default to referencing line segments (generally: geodesics) in any space. That is, regardless of the space, these two cmavo (but not "mi'i") would 'draw' a line from their first argument to their second one.

Note: The endpoints (first and second arguments) of "bi'i" and "bi'o" will be points that are specified via multiple coördinates with respect to a basis. They are not merely scalars. They still must live in the same space (X) and thus must have the same number of coördinates. In this situation, the one-dimensional usage which is defined already outside of this whole proposal merely isomorphically maps scalars which are denoted by "x" and "y" to the their corresponding 1-dimensional point specifications "(x)" and "(y)" respectively. (Notice that, without an additional convention, these will never map to "(x,0,0,...)" and "(y,0,0,...)" respectively, despite the isomorphism that may be established. This is meant to avoid the abusive mixing of notation/spaces: there is no interval from (1,2) to 1, for example. We should always specify that the endpoints are higher-dimensional. This note about mapping 1 to (1) is meant solely for the purpose of making this extension back-compatible and natural.)

This would make the default usage automatically compatible with generalized points (see below). Additionally, line segments are generally useful in geometry of any (nontrivial) dimension, so this functionality would be utile.

This also would allow both "bi'i" and "bi'o" to be defined in any decent space (as opposed to only have "bi'i" be defined, which is the case in the aforementioned subproposal).

In this formulation, "x bi'o y" implies that x is in some sense a starting point of reference/of an imaginary journey and y is the corresponding termination point; both are 'endpoints'/terminals, so to speak. If the space does not 'loop around', then the set produced, however, is still exactly equivalent to that produced by "x bi'i y" and any coloring of the connotations is unmathematical (and, thus, should be avoided in the opinion of lai .krtisfranks.),; the latter is generally preferred.

Note: There is still no established directionality on the linear interval that is produced by "bi'o". However, as long as it does not conflict with any others, we might be able to assume an established order thereupon. "x bi'o y" does mean that "x < y" (along that line).

In this case, we use "ce'ei'oi" (followed by a number larger than 1 if we are being explicit) on either or both points x, y in the constructs "x bi'i y" and "x bi'o y" in order to produce the swept-out higher-dimensional-orthotopal "interval" that was proposed originally. See the 'Handling Generalized Points' section, following, for more details.

"mi'i" will still generalize to a higher-dimensional-ball in the space. (Its functionality, as described previously, and as extended immediately after this section, is unchanged.)

### Proposed Extension A: "mi'i"

"mi'i" can be extended further. Keep the previous definitions and conditions. Now, undefine r. Let <math>r_1, r_2, ..., r_n \geq 0</math>.

Then we can express a new formal tuple <math>r = (r_1, ..., r_n)</math> where the order of the entries correspond to similarly labelled coördinates of points in X with respect to the basis established. Note that r does not live in X; it is just a formal n-tuple which has entries ordered in a corresponding manner - in other words, it is just a list of numbers (scalars in the underlying field, more specifically) with the order of presentation fixed by the basis of X and according to the utterer's intention. Notice that r does not technically change if the basis is changed; in such a situation, it may not be possible to describe the n-dimensional interval in simple terms (using only linear combinations of the entries of the new basis) at all and, in any case, the utterer would generally need to supply an entirely different list <math> r\prime </math> in order to convey the same thought.

- Then we can define "x mi'i r" as <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{N} \cap [1, n]), (d_F (x_i, \alpha_i)</math>
*R*<math>r_i)) \}</math>. Notice that 'd' is now actually '<math>d_F</math>', id est: the metric on the field F. Here, each coordinate of a point <math>\alpha</math> is being compared to the corresponding coordinate of point x; if they are within the specified distance of one another (given by the corresponding entry in the list r), then that coordinate works out; iff all of the coordinates of the point work out, then the point belongs to the interval so described.- This extended form of "mi'i" can be obtained via Cartesian products of linear intervals. We will exploit this fact in the discussion about the endpoint stati (see the section named accordingly).

This essentially returns us to the old situation wherein the interval is no longer an n-ball but an n-cell (matching "bi'i"). The side lengths vary (being <math>2 r_i</math> in length, for each side i). The lines which pass through their corresponding/respective midpoints and which are perpendicular to the corresponding hyperfaces will intersect at a single point, videlicet the first argument of "mi'i" constructs (the 'center'; more appropriately: circumcenter), which is the point from which the various perpendicular distances to the boundaries are each measured (being <math> r_i </math>, for the appropriate/corresponding i).

This definition is good for computer science, graphing, and experimental science. It is almost never used in theoretical mathematics. (Literally never in the experience of lai .krtisfranks., at any rate.)

This additional proposal requires no major update, change, or addition to the glossing/keywords associated with "mi'i" in dictionary definitions, although there would be an implicit understanding of increased generality. If desired, however, "orthotopic interval with given circumcenter" or similar would do nicely.

- Additionally, we could establish the convention-by-definition that: <math>((\exists \rho \geq 0: ((\forall i \in \mathbb{N} \cap [1, n]), (r_i = \rho))) \implies </math> "x mi'i r" = "x mi'i <math>\rho</math>" <math>)</math>; but we would need a way to ensure that the audience recognizes <math>\rho</math> as an n-tuple and not just a scalar. Otherwise, utilization of this convention would be indistinguishable from the previously-mentioned case/proposal wherein the second argument as a single number constitutes the radius of an n-ball.
- This complication can be overcome by mentioning "ce'ei'oi" immediately after "<math>\rho</math>" (list sense) in the "mi'i" construct; if this is done, then we are to understand that "<math>\rho</math>" represents - in short-hand form - a formal tuple of identical entries (each being <math>\rho</math> (in the scalar sense)). The elements of this tuple must never be negative.
- If the utterer explicitly defines/declares <math>\rho</math> to be such a formal tuple, then "ce'ei'oi" is not necessary, although it is also not wrong (and may in fact be helpful and encouraged).

- This complication can be overcome by mentioning "ce'ei'oi" immediately after "<math>\rho</math>" (list sense) in the "mi'i" construct; if this is done, then we are to understand that "<math>\rho</math>" represents - in short-hand form - a formal tuple of identical entries (each being <math>\rho</math> (in the scalar sense)). The elements of this tuple must never be negative.

## Handling Generalized Points

If the input (x and y; the type of one determines the type of the other by forcing it to be the same) are generalized points (such as towns/geographic points), then they likely live in at-least-two-dimensional space, as is the case on Earth. However, with or without Alternative #1, they are being treated as distinct points (assigned real numbers isomorphically if Alternative #1 is not adopted; otherwise, they are free to be points in space). According to this proposal, it is not possible for the interpretation of an interval with these arguments to be anything except one-dimensional. But, for example, maybe we want to discuss the 'rectangular' (more on this immediately later/below) area of the globe between Olympia, Washington, U.S.A. (defined as x) and Tallahassee, Florida, U.S.A. (defined as y). (This rectangle covers much of the continental/contiguous U.S.) "x bi'i y" would, presumably, give the quickest route for a roadtrip or flight from Olympia to Tallahassee (the geodesic line segment (more on this immediately later/below)) or the reverse thereof (from the other to the one); it would not yield the 'rectangle' that we want. The word "ce'ei'oi" fixes this issue. If it is used on a generalized point (rather than a formal tuple - see immediately previous/above ("mi'i" discussion)), then it indicates that that endpoint (and, consequently, all others) is to be treated multi-dimensionally (unless the argument of "ce'ei'oi" is identically and exactly equal to 1). (In order to be clear: in "mi'i", the second argument (the tuple of "radius length(s)") is not considered to be an endpoint for this purpose - or, truly, anywhere within this proposal; it has a distinct nature and possibly typing separate from that of the first argument of "mi'i".) It is not necessary if Alternative #1 is not adopted and if the points are already defined to belong to a well-described space (of known dimensionality) or are decomposed in terms of their basis/represented as a tuple. If we do this, with the former definitions, then "x ce'ei'oi bi'i y" will suddenly indicate not the line(ar interval) between x and y, but instead the 'area between them'. (This area may be visualized thusly: Imagine a map with x and y on it, on a computer. Click on one of these points, drag the cursor to the other. In many programs, a (possibly degenerate) rectangle is swept out, usually with a dashed or dotted outline. The opposite corners of this rectangle are x and y; the remaining corners are given by one coordinate of x and the other coordinate of y, as appropriate. The space highlighted (within this rectangle) is the interval formed.)

Note that "x ce'ei'oi bi'i y" is equivalent to "x ce'ei'oi bi'i y ce'ei'oi" and to "x bi'i y ce'ei'oi". Likewise if we replace "bi'i" with "bi'o".

Note that "ce'ei'oi" has to follow the last argument of "bi'i" and "bi'o" when it is used therein and on the right/later side of BIhI and if the right/later BIhI argument is a scalar of nonnegative integer value; else, it will adopt the BIhI argument as its own (instead), unless its own argument is immediately followed by "boi".

The space in which this interval exists is determined by context. This is a problem even in the unextended (one-dimensional) version of these words in this usage: Is the unordered interval from Paris to London along the Earth's surface (geodesic) or is it a straight line through space (intersecting the Earth's surface at some points as well as the interior)? Is the (un)ordered interval, if geodesic, following the shorter segment of the great-circle connecting the two cities, or the longer one? (The same is true in and with Alternative #1 each.)

Note that in any case x and y need to live in the same space. So, if one is a generalized point, then the other must be. It makes no sense to discuss the interval from 1 to you or from Olympia to my imaginary friend. Moreover, they must have the same dimensionality; placing "ce'ei'oi" on one of them determines the nature of the other (and so is unnecessary); however, the argument of "ce'ei'oi" does need to be compatible with both x and y if present (for "bi'i" and "bi'o"; this is not the case for its being used upon the first argument of "mi'i", but is indeed the case in its being used on the second argument of "mi'i").

## Endpoint Clusivity Stati (Inclusion/Exclusion)

### Notation, Background, Set-Up

Fix a space X which is endowed with a metric d and defined over an ordered field F which is also endowed with a compatible metric <math>d_F</math>; define the dimensionality of X to be dim(X) = n, where n is any nonnegative integer or (for simplicity: countable) infinity; fix a basis B = {<math>e_1, e_2, ..., e_n</math>} thereof. Let x and y live in the same space X. Then there exists <math> x_1, x_2, ..., x_n,\; y_1, y_2, ..., y_n \in F: x = x_1 e_1 +...+ x_n e_n = (x_1, ..., x_n),\; y = y_1 e_1 +...+ y_n e_n = (y_1, ..., y_n)</math>. Let <math>\rho,\; r_1, r_2, ..., r_n \geq 0</math>. Then we can express a new formal tuple <math>r = (r_1, ..., r_n)</math> where the order of the entries correspond to similarly labelled coordinates of points in X with respect to the basis established. Note that r does not live in X; it is just an n-tuple which has entries ordered in a corresponding manner. If n = 1, then we denote: x = (x) = (<math>x_1</math>), y = (y) = (<math>y_1</math>), r = (r) = (<math>r_1</math>).

Assume at least some of the previous proposals.

In this section, "endpoint status" (plural: "endpoint stati") and "clusivity" will refer to the options that an utterer has in specifying whether or not the boundary of the (multidimensional) interval is included xor excluded (along certain hyperplanes/manifolds). If such a boundary is included, then we call it "inclusive", "open", "soft", and "rounded" (due to the notation of using rounded brackets ("(" and ")") or an open circle ("○") in graphical/visual depictions). If such a boundary is excluded, then we call it "exclusive", "closed", "hard", and "square"/"sharp" (due to the notation of using square brackets ("[" and "]") or a closed circle ("●") in graphical/visual depictions). Fix an order '<' on the one-dimensional space(s) to at least one of which every linear interval is taken to belong; in each case of occurrence, appropriately redefine "*R*" <math>\in \{ ``<", ``\leq" \}</math> as determined by intent and context.

A bracket is a symbol which is used in order to denote a piece of the boundary of an interval. It also encodes endpoint status (id est: the clusivity of that part of the boundary). We use rightward-opening brackets for those which, in 1 dimension, bound the space from below/the negative side; these are "(", "[", and "【". "(" denotes such a boundary which is open; "[" denotes such a boundary which is closed; "【" will be used herein as a short-hand way of denoting the general such boundary which may, depending on context, be replaced with "(" and/or "["; this last is called a(n) (opening/rightward-opening) lenticular bracket; its conveyed endpoint status is semantically elliptical in nature. We use leftward-opening brackets for those which, in 1 dimension, bound the space from above/the positive side; these are ")", "]", and "】". ")" denotes such a boundary which is open; "]" denotes such a boundary which is closed; "】" will be used herein as a short-hand way of denoting the general such boundary which may be, depending on context, be replaced with ")" and/or "]"; this last is called a(n) (closing/leftward-opening) lenticular bracket; its conveyed endpoint status is semantically elliptical in nature. If it is not clear (especially with the usage of the lenticular brackets, which can assume various values depending on context and which may or may not be interdependent), brackets which pair or depend upon one another will be subscripted with the same label/index as appropriate.

The status of "【" or of "】" is which of the two options (inclusive or exclusive) it represents in a given instant/context. The status of *R* is which of "<" or "<math>\leq</math>" it represents in a given instant/context.

"<math>\times</math>" denotes the Cartesian product of sets. For sets A and B, <math> A \times B = \{ (a, b):\; a \in A \;\&\; b \in B \}</math>, where (a,b) is a tuple/point with coordinates (namely, in order: a, b) rather than, as the unfortunate notation may suggest, a linear interval from a to b. Note that it is ordered and not commutative. It can be made into a big/serial/iterating operator, denoted: "<math> \times_{i \in C} </math>", where i is an index and it takes every value in a set C subject to the order on C (if there is any).

Denote: <math>\mathbb{N} \cap [1, n] = \boxdot (n)</math>.

This section will use "ce'ei'oi" throughout its body. This is done for the sake of eliminating ambiguity and compatibility with all proposals (and alternatives) herein presented.

Terminology: In "x mi'i r" (for any dimension for x regardless of r) we call x the "center" and r the "radius (length)". Note that, even though it may be called "radius", r really refers to the length in a given dimension. (Radii are really line segments, not the lengths which are associated with them; we abuse terminology here for simplicity.) Any point which has distance from the center (according to the appropriate metric) in a given direction which is exactly equal to the appropriate radius length in the same direction is called an "on-sphere point", "outermost points", or "distant point" of the interval in question. The set of all on-sphere points of an interval is its "outer boundary" or "(outer/minimal enclosing) sphere". Note that "mi'i" may actually generate an n-orthotope (per Proposed Extension A) and thus may not seem or be spherical; the terminology does not vary in order to explicitly and individually account for such cases. The center of a "mi'i" interval is an element of the boundary (which is, generally, partitioned into outer and inner parts; the center is the sole element of the inner part thereof); thus, the center is an endpoint with a clusivity status.

Terminology: Each (n-1)-side of the n-orthotope produced via interval constructs will be called a "boundary-part" (these are the aforementioned hyperplanes segments); these generalize the notion of "endpoints" (in a slightly different way from how the arguments of BIhI do so) from the one-dimensional case. They may still be called "endpoints" in this article, even though they are not actually points. For a given interval, the set of all such boundary-parts thereof is the boundary of the interval. "mi'i" is taken to have a whole and indivisible outer boundary (without individual boundary-parts) when its dimensionality is greater than 1; when its dimensionality is equal to 1, the two outermost (end)points may be taken to each constitute a boundary-part of the outer boundary; there is an additional (inner) boundary-part, which is exactly the singleton set of the center point.

'**!'** **Important Result**: The n-orthotope formed by "x ce'ei'oi bi'i y (ce'ei'oi)" is <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X:\; ((\forall i \in \boxdot (n)),\; (x_i</math> *R* <math>_{-1,i} \alpha_i </math> *R* <math>_{1,i} y_i)) \} = \times_{i \in \boxdot (n)} \;(</math>【<math>_{i}\; x_i,\; y_i </math>】<math>_{i} \;)</math>, where: <math>\forall i \in \boxdot (n)</math>, the status of '【<math>_{i}</math>' is determined by/isomorphic/logical equivalence with/to the status of ' *R*<math>_{-1,i}</math>' and likewise the status of '】<math>_{i}</math>' is analogously determined by/isomorphic/logical equivalence with/to the status of ' *R*<math>_{1,i}</math>'.

### Current Situation: the One-Dimensional Case

This section is according to (an interpretation of) the CLL.

In one dimension, we can specify the status of either endpoint of the (linear) interval via use of members of GAhO immediately next to the member of BIhI being used, on the appropriate side. For example, "li no ga'o bi'o ke'i li pa" represents the interval [0, 1). Notice that "ga'o" (which specifies an inclusive endpoint/boundary-part on the negative side) is on the leftern/first-uttered side of "bi'o" (id est: immediately preceding it) and follows "li no"; this means that the lesser endpoint (in this case, the point 0) is to be included. Likewise, "ke'i" (which specifies an exclusive endpoint/boundary-part) is on the rightern/last-uttered side of "bi'o" (id est: immediately following it) and precedes "li pa"; this means that the greater endpoint (in this case, the point 1) is to be excluded.

The exact same is true if we replace "bi'o" with "bi'i", although we will lose the implication that <math>0 \leq 1</math>.

Members of GAhO, at least in BIhI constructs, seem to be used - if at all - only adjacent to the member of BIhI itself. lai .krtisfranks. does not know if they are used in any other context or in any other way. There seems, to lai .krtisfranks, to be no recognition nor convention concerning how to interpret the absence of any explicit members of GAhO on one or both sides of BIhI. It does not seem possible, to lai .krtisfranks., for multiple members of GAhO or of BIhI to consecutively succeed one another (of the same selma'o) whilst being 'naked' (id est: with their dictionary meaning, without being quoted or deleted, without being JE- or JOI-connected, etc).

The structure of a BIhI construct is: [endpoint_{1}] (GAhO*) BIhI* (GAhO*) [endpoint_{2}].

### Assumption/Proposal anent Unmentioned GAhO

The content of this section may count as an additional proposal. In any case, it is assumed throughout the rest of the article.

If, between an argument of BIhI and the BIhI cmavo itself, there is no explicitly mentioned member of GAhO, then one is still assumed to be present and to apply to and only to the appropriate boundary-part; the clusivity content of an implicit GAhO cmavo, such as in this case, is elliptical/unspecified/vague - it may or may not be determined by context, practicality, or intent, or it may be unimportant (thus, possibly, representative of either possibility).

For your information: "xau'u'oi" is a member of GAhO which is elliptical with respect to clusivity (status), as described immediately previously. It is proposed separate from and independent of/to this whole proposal or any of its parts. However, the two complement eachother well.

Whether implicit (as described herein) or explicitly elliptical, such an elliptical clusivity status will be denoted by lenticular brackets in linear intervals.

It does not make sense for a nontrivial interval of dimensionality at least 1 to neither include nor exclude any given boundary-part in the current framework of what is easily supported and expressible by Lojban; even mathematically and independent of the language, such a statement is true if it is restricted to the one-dimensional case (with end-**points**). Therefore, we must assume that one xor the other case applies. Thus, it should not be contentious that this assumption is made.

### Proposed Extension B: GAhO with "mi'i", a simple case

There appears to be no prior usage of GAhO with "mi'i" except as proposed here, a notion which lai .krtisfranks. separately formulated in his considerations for this article. The suggestion there is as follows:

GAhO before "mi'i" indicates the "endpoint status" of the center of the interval. "ga'o" there indicates that the center (the entire internal boundary-part) is definitely included (thereby forming a continuous/contiguous linear interval); the radius-distanced endpoints may or may not be included. "ke'i" there indicates that the center is definitely excluded (thereby forming a broken/punctured linear interval which misses only its center and possibly the radius-distanced endpoints). This usage in nondegenerate cases has no bearing on the clusivity of the radius-distanced endpoints (unless one specifies explicitly that these stati are mutually determinable).

GAhO after "mi'i" indicates the endpoint status of any point which has distance from the center exactly equal to the radius length specified. "ga'o" here indicates that all of the on-sphere points of the interval are definitely included (causing the outer boundary of the interval to be included; this is pictorially represented by a solid boundary curve). "ke'i" here indicates that all of the on-sphere points of the interval are definitely excluded (causing the outer boundary of the interval to be excluded; this is pictorially represented a dashed/dotted/broken boundary curve). Notice that every on-sphere point shares the same status as every other on-sphere point of a given interval. In nondegenerate cases, this usage has no bearing on the clusivity status of the center of the specified interval (unless one specifies explicitly that these stati are mutually determinable).

These descriptions apply for the one-dimensional case, but generalize - with little change - to arbritrary dimensionality. The only difference is that the interval generated will be a possibly-punctured line (segment), disc, ball, glome interior, or - generally - n-ball, depending on the dimensionality.

Let <math> x \in \mathbb{R}</math>. So, <math> x \leftrightarrow (x)</math>. Recall: <math>\rho \geq 0</math>. Also, 'd' denotes the distance in this space; note that it is positive-definite. These are all one-dimensional:

- "x mi'i <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0\; </math>
*R*<math>\; d(x,z)\; </math>*R*<math>\; \rho \}</math>.

- "x ga'o mi'i <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0\; \leq\; d(x,z)\; </math>
*R*<math>\; \rho \}</math>.

- "x ke'i mi'i <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0\; <\; d(x,z)\; </math>
*R*<math>\; \rho \}</math>.

- "x mi'i ga'o <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0\; </math>
*R*<math>\; d(x,z)\; \leq\; \rho \}</math>.

- "x mi'i ke'i <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0\; </math>
*R*<math>\; d(x,z)\; <\; \rho \}</math>.

- "x ga'o mi'i ga'o <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0 \leq d(x,z) \leq \rho \}</math>.

- "x ga'o mi'i ke'i <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0 \leq d(x,z) < \rho \}</math>.

- "x ke'i mi'i ga'o <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0 < d(x,z) \leq \rho \}</math>.

- "x ke'i mi'i ke'i <math>\rho</math>" means: <math>\{ z \in \mathbb{R} :\; 0 < d(x,z) < \rho \}</math>.

_

For emphasis: Even though the forms of the mathematical expressions are approximately the same (similar on first glance) to the analogous expressions for "bi'o", and despite the formal similarities in the Lojban utterances, it is important to realize that <math>\rho</math> here is **not** an endpoint of the linear interval. The presence of 'd' in the definition makes a world of difference.

### Main Proposal #2

Recall: dim(X) = n.

Note: In this section, it is taken to be the case that two mutually adjacent members of GAhO do not compound to form a member of GAhO* (in/as which they cannot be separated).

It would be nice to be able to specify the status of boundary-parts of higher-dimensional intervals. But there is a great deal of customizability that is available. The pair "vau'e'oi" and (its elidable terminator) "vau'o'oi" handle this.

The way that they work is as follows:

- "vau'e'oi" opens a scope; this is called the "interval brackets scope" (hereby named "(the) IBS"). The IBS is closed via explicit use of "vau'o'oi", or immediately upon the utterance of any word which is not a member of selma'o GAhO*, or immediately upon the event of the number of explicitly utterred GAhO* members exceeding n. Thus, the only words which may belong within the IBS are members of GAhO* and they number at most n. Any GAhO cmavo explicitly used immediately after the nth explicitly used one in an IBS is ignored.
- The IBS has n ordered slots; these are called "IBS slots" (or, here, just "slots"). Initially, no slots are filled. For each IBS slot, a single explicitly utterred member of GAhO* can fill it. Once one slot is filled, the next explicitly utterred GAhO* member fills the next available slot if possible/it is available. The first explicitly utterred GAhO* member fills the first IBS slot. Members of GAhO* do not need to be separated nor connected from one another in any way. They are Quine-quoted and treated as separate entities
**automatically**. Thus, no explicit quotes are necessary. - If the IBS is closed 'prematurely' (id est: before n GAhO* members are explicitly utterred), then any IBS slot which is not explicitly filled at this point is taken to be filled by a vague/elliptical value.
- Syntactically, the IBS result is treated as a single occurrence of GAhO*. Thus the IBS construct must be adjacent to an explicit member of BIhI.
- Semantically/practically, the result is a formal and ordered tuple of interval brackets, the elements of which are the brackets supplied to the IBS in order such that the <math>i</math>th element of the formal tuple produced being the bracket supplied to the <math>i</math>th IBS slot. The location of this result relative to the adjacent BIhI member forces the nature of these brackets (to be rightward-opening iff the result is utterred before/is to the left of the adjacent BIhI member; to be leftward-opening iff the result is utterred after/is to the right of the adjacent BIhI member). Denote this result to be <math>(</math>
*B*<math>_1, ...,</math>*B*<math>_n)</math>. Notice that this usage of the character "B" is not the same as elsewhere in this article (where it, for example, may have meant a particar basis); here, it is an individual bracket with a defined clusivity status. The role of this bracket is determined by the location of its IBS with respect to the relevant BIhI; its particular realization with respect to that boundary - that is, its applicability to one or more boundary-parts in that boundary - is determined by the slot which it fills and the basis determined by context. - The <math>i</math>th member of this tuple (namely,
*B*<math>_i</math>) acts as the corresponding bracket (opening or closing, open or closed) for the <math>i</math>th (component-)interval (parallel to the <math>i</math>th basis element) in the Cartesian product reëxpression/decomposition of the overall interval if such is possible. In other words, the clusivity of the ith hyperplane is that of the bracket B_{i}. The order is important and corresponds via index.

Reiterating: An IBS of n > 0 slots that is closed while having only m slots explicitly filled by GAhO*, where 0 < m < n and m is an integer, has the last n-m slots filled implicitly by an elliptical GAhO* element (with restrictions intuitively/implicitly placed upon it due to context). Notice that GAhO cmavo connected by JE or JOI, for example, form a single element of GAhO* for the purpose of filling slots. There is no separator between filled slots; the GAhO* elements may just be rattled off, separated automatically and filling the slots successively.

- Example: Consider the interval <math>I</math> generated/named by "<math>(x_1, x_2, ..., x_n)</math> ce'ei'oi (<math>n</math>) vau'e'oi 【<math>_{1}</math> 【<math>_{2}</math> ... 【<math>_{n}</math> (vau'o'oi) bi'i vau'e'oi 】<math>_{1}</math> 】<math>_{2}</math> ... 】<math>_{n}\;\;</math> (vau'o'oi) <math>(y_1, y_2, ..., y_n)</math> (ce'ei'oi (<math>n</math>))", where <math>(x_1, x_2, ..., x_n)</math> and <math>(y_1, y_2, ..., y_n)</math> are expressed with respect to the basis of the space.
- Notice the lack of commas/separators between brackets. Also note that only one occurrence of "ce'ei'oi" is necessary in this case (since we are working with "bi'i"). The usage of "vau'o'oi" here (in both cases) is optional.
- Then <math>I = \{ \alpha = (\alpha_1, ..., \alpha_n) \in X:\; ((\forall i \in \boxdot (n)),\; (\alpha_i \in </math>【<math>_i\; x_i,\; y_i </math>】<math>_i)) \} = \times_{i \in \boxdot (n)} \;(</math>【<math>_{i}\; x_i,\; y_i </math>】<math>_{i} \;)</math>.
- Thus, we see that the <math>i</math>th opening bracket goes with <math>x_i</math> and the <math>i</math>th closing bracket goes with <math>y_i</math>.
- Even though it has the same generic form as was expressed in the final part of the previous "Current Functionality" subsection, the meaning is rather different. There, the brackets were elliptical/vague/generic/general. Here, they are specified by the utterrer to be whatsoever was said in the IBS.

- Example: Consider <math>I</math> generated/named by "<math>(x_1, x_2, x_3)</math> ce'ei'oi (ci) vau'e'oi ga'o ke'i ga'o (vau'o'oi) bi'i vau'e'oi ke'i ke'i ga'o (vau'o'oi) <math>(y_1, y_2, y_3)</math> (ce'ei'oi (ci)).
- Then <math>I = [x_1 ,\; y_1) \times (x_2 ,\; y_2) \times [x_3 ,\; y_3]</math>. This is the basically the referent of "vei <math>x_1</math> ga'o bi'i ke'i <math>y_1</math> ve'o pi'u vei <math>x_2</math> ke'i bi'i ke'i <math>y_2</math> ve'o pi'u vei <math>x_3</math> ga'o bi'i ga'o <math>y_3</math> ve'o".

Note that in the precious two examples, so order was imposed on each linear subspace of X. This can be done, mathematically. But it was done here for convenience/for the sake of having established notation in order to make sense of the meaning in a rigorous way. If such order exists on a given linear subspace of X, then all is well for that particular instance of its imposition. However, when interpreting, one must remember to "unimpose"/"unassume" such an ordering. The linear intervals are just specified by two endpoints; the endpoints themselves can be switched (especially systematically). This, however, is a reasonable and semi-canonical way to represent the linear intervals (as long as one remembers that the endpoints need no be in the specified order); it would be confusing to switch only some of them.

Moral of the story: The IBS causes the brackets that it is given to be sent off to couple with the coordinate to which they belong, in order.

Postliminarily: It is not grammatical to have naked members of GAhO uttered consecutively in a BIhI construct except when they all are in an IBS (as enclosed by "vau'e'oi" and (possibly elided) "vau'o'oi").

### Further Proposal: a Nice and Simple Case for Brackets

The content of this section supposes Main Proposal #2 as being accepted and applicable.

In this section, back-compatibility, utility/ease, and a simple and nice case are handled.

What if we do not want to say all of that? How do we support back-compatibility this way? What if all of the brackets submitted is a certain IBS are the same? Well, here is some more functionality: If the last case holds and if "ce'ei'oi" is mentioned so as to apply to an endpoint (rather than a radius length(s) entry), then we can dispense with the IBS altogether. In the multidimensional case, a single bracket *B* on a given side of "bi'i" is taken to be isomorphic with the formal ordered interval-bracket tuple (*B*, ..., *B*), where the number of entries therein is the dimensionality of the space n. Id est: each GAhO cmavo actually constitutes an n-slot IBS such that every entry in it is, explicitly, that very same cmavo. Therefore, "vau'e'oi vau'o'oi" is equivalent to a single instance of "xau'u'oi" alone, allowing for n-dimensionality; this in particular is back-compatible with earlier claims/assumptions/proposals.

In the case of all the brackets on one side being (of) the same (status), we can just call the tuple by the name of the shared bracket (status). If we are working in an n-dimensional space and we are given only one bracket B outside of an IBS on a given side of "bi'i" explicitly, then we know that the IBS actually should should actually contain n brackets of form B (of the same status). This can work simultaneously on both sides of BIhI.

In short: "ce'ei'oi n ga'o bi'i" is equivalent to "ce'ei'oi n vau'e'oi ga'o ga'o [...] ga'o vau'o'oi bi'i", where "ga'o" is explicitly mentioned n times in the latter sentence. Analogy/substitution for other cases (such as: using "ke'i", having the brackets on the other side of the BIhI cmavo, or using another cmavo in BIhI) follows by the syntactic uniformity within selma'o.

Thus, <math>\forall x, y \in \mathbb{R}</math>, "x ga'o bi'i y" is mapped isomorphically to "x (ce'ei'oi pa) vau'e'oi ga'o (vau'o'oi) bi'i (vau'e'oi (vau'o'oi)) y". Recall, of course, that "x" here actually maps to "(x)" isomorphically too, and likewise for "y"; this detail was not shown for clarity, since parentheticals were employed in that example/string in order to represent optional utterrances, and the endpoints are definitely not optional in general.

Further: At most one naked member of GAhO which is not enclosed by an IBS xor at most one IBS may be explicitly used on a given side of BIhI; any more of either and/or the use of both together is ungrammatical.

### Proposed Extension B'

This section assumes adoption of Proposed Extension B. If anything is to be done with "mi'i" (even in the 1-dimensional case) this subproposal (respectively Proposed Extension B) needs to be accepted.

In "mi'i", the first bracket never needs to be processed through an IBS. Either the center point is included or excluded, and this can be handled one-dimensionally (id est: by just using a member of GAhO on its own). Only one first bracket can be used/appear.

If "mi'i" is being used in order to produce/describe an n-ball (rather than an n-orthotope, per Proposed Extension A), then only one second (formally "closing") bracket can be used/appear, and- thus- it need not be processed through an IBS. In this case, every on-sphere point is either included xor they are all excluded; specification of this status can be handled one-dimensionally (id est: by just using a member of GAhO on its own).

Let <math> x \in X</math>. So, <math> x = (x_1, ..., x_n)</math>. Recall: <math>\rho \geq 0</math>. Also, 'd' denotes the distance in this space; note that it is positive-definite. "R" is used, as before, for an inequality relation that may be either strict or loose (the latter admits the possibility for equality). These are all n-dimensional:

- "x (ce'ei'oi (n)) mi'i <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0\; </math>
*R*<math>\; d(x,z)\; </math>*R*<math>\; \rho \}</math>.- This is a possibly punctured disc/ball which may or may not include its external boundary (but whatsoever it does there, it does so to the totality thereof).

- "x (ce'ei'oi (n)) ga'o mi'i <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0\; \leq\; d(x,z)\; </math>
*R*<math>\; \rho \}</math>.- This is an unpunctured disc/ball; it may or may not be united with its sphere (boundary); it would be drawn with shading throughout the interior.

- "x (ce'ei'oi (n)) ke'i mi'i <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0\; <\; d(x,z)\; </math>
*R*<math>\; \rho \}</math>.- This is a punctured disc/ball; it may or may not be united with its sphere (boundary); it would be drawn with shading throughout the interior except at the center, where a small open and unshaded ball would be drawn/notated.

- "x (ce'ei'oi (n)) mi'i ga'o <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0\; </math>
*R*<math>\; d(x,z)\; \leq\; \rho \}</math>.- This is a possibly punctured disc/ball that is definitely united with its sphere (boundary); it would be drawn with a solid surface on the outside.

- "x (ce'ei'oi (n)) mi'i ke'i <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0\; </math>
*R*<math>\; d(x,z)\; <\; \rho \}</math>.- This is a possibly punctured disc/ball that is definitely not united with its sphere (boundary); it would be drawn with a broken/dashed/dotted surface on the outside.

- "x (ce'ei'oi (n)) ga'o mi'i ga'o <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0 \leq d(x,z) \leq \rho \}</math>.
- This is a closed (solid) unpunctured disc/ball; it definitely includes both its center and its external boundary; it would be drawn with a solid surface on the outside and shading throughout the interior.

- "x (ce'ei'oi (n)) ga'o mi'i ke'i <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0 \leq d(x,z) < \rho \}</math>.
- This is an open unpunctured disc/ball; it definitely includes its center but definitely excludes it external boundary; it would be drawn with a broken/dashed/dotted surface but with shading throughout the interior.
- This is the typical definition of "ball" (or "disc" in two dimensions). It has only interior points, but it has all of them.

- "x (ce'ei'oi (n)) ke'i mi'i ga'o <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0 < d(x,z) \leq \rho \}</math>.
- This is a punctured disc/ball which is united with its external sphere (external boundary); it definitely includes its external boundary but definitely excludes its center; it would be drawn with a solid surface and shading throughout the interior except at the center, where a small open and unshaded ball would be drawn/notated.

- "x (ce'ei'oi (n)) ke'i mi'i ke'i <math>\rho</math>" means: <math>\{ z = (z_1, ..., z_n) \in X :\; 0 < d(x,z) < \rho \}</math>.
- This is a punctured disc/ball which excludes its external sphere (external boundary); it definitely excludes both its external boundary and its center; it would be drawn with a broken/dashed/dotted surface and shading throughout the interior except at the center, where a small open and unshaded ball would be drawn/notated.
- This is the typical definition of a punctured ball (disc in two dimensions). It is useful for the definition of limit points in topology.

Clearly, as before, the definitions extend exceptionally easily for this interpretation of "mi'i". In the mathematics, the only formal change in the condition was replacing each "<math>\mathbb{R}</math>" with <math>X</math>. Of course, the points themselves have multiple coordinates and thus the meaning of 'd' gets slightly more complicated, but that is all under the hood (determined by the definition of 'd' and/or intuition about what would be appropriate for the space, given X itself).

### Proposed Extension C

This section assumes adoption of Proposed Extension A and Proposed Extension B' (and thus Proposed Extension B).

If working with "mi'i" where the first argument <math>x = (x_1, ..., x_n)</math> lives in n-dimensional space and if the second argument <math>r = (r_1, ..., r_n)</math> is a formal ordered tuple of n nonnegative extended-real numbers, then an n-cell is produced, in which case "vau'e'oi" and "vau'o'oi" work exactly as before (with "bi'i"). But it is important to note that the IBS can be nontrivial only on the second-argument side of "mi'i".

Throughout this section, "d" still denotes a distance function on the space X.

- Example: Consider the interval <math>I</math> generated/named by "<math>(x_1, x_2, ..., x_n)</math> (ce'ei'oi (<math>n</math>)) 【 mi'i vau'e'oi 】<math>_{1}</math> 】<math>_{2}</math> ... 】<math>_{n}\;\;</math> (vau'o'oi) <math>(r_1, r_2, ..., r_n)</math> ce'ei'oi (<math>n</math>)", where <math>(x_1, x_2, ..., x_n)</math> is expressed with respect to the basis of the space (and represents a point therein) and <math>r</math> formally corresponds via indices coordinatewise.
- Notice that the second instance of "ce'ei'oi" is necessary since we
**must**specify that we have a formal ordered tuple of radii lengths. In neither case is explicit mention of n necessary as an argument of "ce'ei'oi", provided that the audience knows the size of the tuple r and/or the dimensionality of the space X (which is to say the number of coordinates of x; all of these must match).- Of course, if the tuple r is specified explicitly, especially entry-by-entry, or if everyone knows that it is a formal ordered nontrivial tuple, then "ce'ei'oi" is not necessary. Additionally, if r is trivial (in the sense of having at most one entry), then "ce'ei'oi" is elidable by convention.

- Notice that the first-uttered/leftern bracket is alone.
- If, here, '【' = '(' here, let <math>C = \{ (x_1 , x_2 , x_3) \} </math>, id est the singleton of the central point; if, here, '【' = '[', then let <math> C = \{ \} </math>. Then <math>I = \{ \alpha = (\alpha_1, ..., \alpha_n) \in X:\; ((\forall i \in \boxdot (n)),\; ( d(x_i, \alpha_i) \in [0,\; r_i </math>】<math>_i)) \} \setminus C</math>.
- Notice that it is
: <math>I = \{ \alpha = (\alpha_1, ..., \alpha_n) \in X:\; ((\forall i \in \boxdot (n)),\; ( d(x_i, \alpha_i) \in </math>【 <math>0,\; r_i </math>】<math>_i)) \}</math>.**NOT**

- Notice that it is

- Notice that the second instance of "ce'ei'oi" is necessary since we

- Example: Consider <math>I</math> generated/named by "<math>(x_1, x_2, x_3)</math> (ce'ei'oi (ci)) (vau'e'oi) ga'o (vau'o'oi) mi'i vau'e'oi ke'i ke'i ga'o (vau'o'oi) <math>(r_1, r_2, r_3)</math> ce'ei'oi (ci).
- Then <math>I = (x_1 - r_1 ,\; x_1 + r_1) \times (x_2 - r_2 ,\; x_2 + r_2) \times [x_3 - r_3 ,\; x_3 + r_3]</math>.
- Notice that, despite the inclusion status of the first argument, the leftern brackets in the Cartesian expression of this orthotope have varied clusivity status. The important thing is that the intervals are continuous (not missing their central point x
_{i}) due to the explicit presence of "ga'o" immediately before "mi'i"; the clusivity stati of the leftern bracket and of the rightern bracket in each Cartesian-productand should mutually match - and they are determined determined by the member of GAhO in the corresponding ISB slot in the ISB immediately following "mi'i".

- Notice that, despite the inclusion status of the first argument, the leftern brackets in the Cartesian expression of this orthotope have varied clusivity status. The important thing is that the intervals are continuous (not missing their central point x

- Then <math>I = (x_1 - r_1 ,\; x_1 + r_1) \times (x_2 - r_2 ,\; x_2 + r_2) \times [x_3 - r_3 ,\; x_3 + r_3]</math>.
- Example: Consider <math>I</math> generated/named by "<math>(x_1, x_2, x_3)</math> (ce'ei'oi (ci)) (vau'e'oi) ke'i (vau'o'oi) mi'i vau'e'oi ke'i ke'i ga'o (vau'o'oi) <math>(r_1, r_2, r_3)</math> ce'ei'oi (ci).
- Then <math>I = \big( (x_1 - r_1 ,\; x_1 + r_1) \times (x_2 - r_2 ,\; x_2 + r_2) \times [x_3 - r_3 ,\; x_3 + r_3] \big) \setminus (x_1 , x_2 , x_3 ) </math>.
- Notice that it is
: <math>I = ((x_1 - r_1 ,\; x_1) \cup (x_1 ,\;x_1 + r_1)) \times ((x_2 - r_2 ,\; x_2) \cup (x_2 ,\;x_2 + r_2)) \times ([x_3 - r_3 ,\; x_3) \cup (x_3 ,\;x_3 + r_3])</math>.**NOT**

- Notice that it is

- Then <math>I = \big( (x_1 - r_1 ,\; x_1 + r_1) \times (x_2 - r_2 ,\; x_2 + r_2) \times [x_3 - r_3 ,\; x_3 + r_3] \big) \setminus (x_1 , x_2 , x_3 ) </math>.

## Vocabulary/Semantics that have been Introduced

Mostly, old vocabulary has been expanded in functionality.

**Old/CLL vocabulary with expanded functionality**:

- mi'i - Originally, this word denotes/construct a line or line segment which was (taken to be) centered on the point specified by the first argument and which had length specified by the second argument. Main Proposal #1 allows "mi'i" to denote/construct a volume in n dimensions which is bounded by a sphere which is centered at the point specified by the first argument and which is of the radius specified by the second argument. Proposed Extension A allows "mi'i" to denote/construct a volume enclosed by an n-orthotope with axes parallel to those of the space (the basis) such that each side has a distance from the point, specified by the first argument, which is equal to the corresponding entry in the list which is specified by the second argument; if all entries in the list which is the second argument are the same, then the second argument may be represented by just that number via the use of "ce'ei'oi" (see later).
- bi'i - Originally, this word denotes/constructs a line, linear ray, or line segment in one-dimensional space between two specified points (rays have one such point being infinite; lines have both being infinite). Main Proposal #1 extends the functionality so that this word denotes/constructs a volume enclosed by an n-orthotope in n-dimensional space (which may have any, all, or none of its faces/sides at infinity), such that each axis thereof is parallel to one axis of the space (defined by a single element of the basis); the two arguments of "bi'i" specify some pair of mutually opposite vertices on the n-orthotope - there coordinates with respect to the relevant basis determine the corresponding sides and are the corresponding endpoints in an expression of the generated volume by Cartesian product of one-dimensional linear intervals (which may be given by CLL/old "bi'i" itself).
- bi'o - Originally, this word acted as "bi'i" does with the further assumption that the first argument is lesser/the origin and the second argument is greater/the destination. Its functionality was not really generalized in this whole proposal.
- ga'o/ke'i - If every bracket in an IBS (see "vau'e'oi", later) would be explicitly specified as a single particular member of GAhO, then the whole IBS may be abbreviated to/represented by a single instance of the same unenclosed in an IBS.

For all of these, if the points specified are one-dimensional, generic/geographic, or cannot be specified in terms of a tuple of coordinates (as given by the basis of the space), then they may be represented by a simple point/name (not enclosed in a tuple), which is isomorphic to a one-tuple of only that entry.

**New vocabulary**:

- ce'ei'oi
- vau'e'oi
- vau'o'oi
- IBS

**Independent vocabulary**:

- xau'u'oi

## Miscellany

"mi'i" is really good for error bars in the sciences. In English, scientists often say stuff like "g is 9.85 plus-or-minus .05 meters per second per second". This is abusive. The only options for the value of g in such a case would be 9.80 m/s^{2} or 9.90 m/s^{2}. What they mean is that the value of g is between these two values (possibly including either of them). Following the format of the example quote, we have "mi'i" being the intention, with 9.85 functioning as the center and .05 as the radius. In describing a data set, one should make sure to say that the variable (usually dependent) *belongs* to this set, rather than it *being* this set. This variable will be the one with error bars in the graph. If a single variable is described in such a manner, the error bars graphically are parallel to only one axis: the actual value (as measured, within standard deviation/error) can shift in this direction (so long as it remains within the bars) but cannot shift in any other. If the error bars are applied to the *point*, rather than the *variable* (which acts as a coordinate of the former), though, then the error bar will (under this proposal) envelop the point in a ball of the given radius; the actual value (as measured, within standard deviation/error) could thus shift in any direction within n-space so long as it remains within the provided radius of the given (measured, central) value. Adopting Proposed Extension A, each coordinate can be individually and independently assigned/associated with an error; the error bars will graphically be parallel to each axis (or will be 0); the actual value (as measured, within standard deviation/error for each measurement/variable/coordinate) can shift relative to the data point along each axis so long as it stays within the axis-appropriate radius of the data point.

### Further Ideas

- It may be beneficial to us to figure out a system by which to generate basic shapes such a convex hulls via specification of important points (or sets of points). The current framework does not support such generalization since only two arguments of BIhI are presently allowed.

- We may want a separate word for or modifier of "mi'i", here denoted by "
", such that: for <math>I \prime</math> being the interval generated/named by "<math>(x_1, x_2, ..., x_n)</math> (ce'ei'oi (<math>n</math>)) 【**NTRVL**vau'e'oi 】<math>_{1}</math> 】<math>_{2}</math> ... 】<math>_{n}\;\;</math> (vau'o'oi) <math>(r_1, r_2, ..., r_n)</math> ce'ei'oi (<math>n</math>)", where <math>x = (x_1, x_2, ..., x_n) \in X</math> is expressed with respect to the basis of the space <math>X</math> (and represents a point therein) and <math>r = (r_1, r_2, ..., r_n)</math> formally corresponds via indices coordinatewise, then <math>I \prime = \{ \alpha = (\alpha_1, ..., \alpha_n) \in X:\; ((\forall i \in \boxdot (n)),\; ( d(x_i, \alpha_i) \in </math>【 <math>0,\; r_i </math>】<math>_i)) \}</math>.**NTRVL**- Notice that the second instance of "ce'ei'oi" is necessary since we
**must**specify that we have a formal ordered tuple of radii lengths. In neither case is explicit mention of n necessary as an argument of "ce'ei'oi", provided that the audience knows the size of the tuple r and/or the dimensionality of the space X (which is to say the number of coordinates of x; all of these must match).- Of course, if the tuple r is specified explicitly, especially entry-by-entry, or if everyone knows that it is a formal ordered nontrivial tuple, then "ce'ei'oi" is not necessary. Additionally, if r is trivial (in the sense of having at most one entry), then "ce'ei'oi" is elidable by convention.

- Notice that the first-uttered/leftern bracket is alone.

- Notice that the second instance of "ce'ei'oi" is necessary since we