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Current Functionality: - spacing
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).

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