non-logical interval connective: ordered interval with specified endpoint/terminus x1 and signed measure/length/duration x2; interval between x1 and x1 + x according to the ordering of the space.
This word is a member of selma'o BIhI. The space must be ordered or directed in some sense. Let the space X be the traditionally-ordered field of real numbers, and x and y be in X (or x be a geometric point which is represented in such manner and which belongs to a specified line/ray which has a defined positive direction and a specification of the meaning of the length of 1 unit); then, "x bi'oi y" represents the interval between x and x + y, where y is signed. In other words, this interval is like the body of the string in the scenario in which one of its endpoints is tacked down at x and the other is dragged out a distance y therefrom in the directions specified by the signum of y (with the string being pulled taut and being of length exactly y). Maintaining the same notation and continuing discussion of "x bi'oi y": if the clusivity for the first term (or 'on the left') is specified, then x is or is not included in the referenced interval as specified; if the clusivity for the second term (or 'on the right') is specified, then x + y is or is not included in the referenced interval as specified (note: this does not generally indicate whether y is included in the referenced interval). Units, particularly for y, must be explicit when appropriate. In a connected acyclic directed graph, x would be a node and y would be the graph-geodesic distance therefrom in the direction (if y is positive), or in the opposite direction (if y is negative), of the graph. This definition enables the ISO 8601 standard for time interval specification in the cases (1) of a starting time/pastward endpoint (namely x) and a duration (y: y > 0), and (2) of an ending time/futureward endpoint (namely x) and a duration (y: y < 0), where notation for "x" and "y" (in the usage "x bi'oi y") is continued from the earlier discussion in this note in the case of the number lines or the real; note that, in this usage case, x is a point in time (a timestamp) and y is a duration (elapsed time or measure of time, and which thus has explicitly specified units or something akin to the ISO 8601 standard notation for durations). See this proposal for more information on related topics: https://mw.lojban.org/papri/Proposal:_loi_lerfu_tcita_detri;_the_final_word_on_the_problem_of_dates_and_times%3F . Importantly, the ISO 8601 standard in the case of an ending time and a duration has the terms expressed in the order of y followed by x with y being positive; when this word is used for such purposes, y must be negative, but the order of x and y may be exchanged via usage of "se" (resulting in "sebi'oi"). Generally, the distinction between points/'positions' (represented by x) and measures/vectors/'displacements'/'translation vectors' (represented by y), in the sense of affine spaces, must be maintained when using this word. See also: "bi'o", "bi'i", "mi'i", "bi'oi'au", and https://mw-live.lojban.org/papri/Extended_Dimensionality_of_Interval_cmavo .