xau'o'o
experimental cmavo mekso convention cancellation This word is followed by an evaluated li clause (yielding specifically a number n, which probably is a nonnegative integer, or a set thereof (the elements of which will also be denoted by simply n, for simplicity) ). It repeals the nth most recent/last convention (counting backward) for mekso/mathematical expressions defined by xau'e'o. In order to repeal all such conventions (returning to the conventions specified by some grammar collection), the set should include all integers less than or equal to some sufficiently large integer (which might be ro); ro denotes the first rule defined (the last rule counting backward from the most recent) it does not reference "all rules defined"; the most recent rule is denoted by pa. Any named body/ordered collection of rules (such as ".pemas.") is treated as being only one rule for the purposes of such counting.


zoi'ai
experimental cmavo nonmekso quote/name substitution for ordered collection of prescriptions, descriptions, definitions, etc. Delimited nonmath/nonmekso quote (works like zoi in this respect). Treats the quote as a substitute for some formal collection of rules or mathematical description/definitions/notations; the exact meaning of the quote must be inferred as is the case with la or any quote; the quote is treated as a single block of text representative a single entity so described in only that case/context by the utterer; can be used to clarify the interpretation of text (convention specification; text need not be mathematical (in which case, it must be used in a metalinguistic scope)) or as an operand of certain mathematical operators (or, more generally perhaps, bridi). Might be useful for quoting names as descriptors for arguments of operands. For example: orderings, metrics, bases, densities, analytic properties, conventions, etc. can all be more easily described by a moniker than by a formal mathematical description. See also: ju'au, se'au, mau'au.


cu'au'ei
experimental cmavo mekso binary/unary operator: multinomial coefficient/binomial coefficient/choose If unary, a list (n_{1},...,n_r) must be supplied as input; the terms of the list may not be distinct; for appropriate values, output is ((n_{1} +...+ n_r)!)/((n_{1})! *...* (n_r)!). If binary, a single number n must be supplied for the first variable and the second variable must be a list (k_{1},...,k_s); for appropriate values, the output is (n!)/((k_{1} !) * ((n  k_{1})!) *...* (k_s !) * ((n  k_s)!)); note that this is not exactly the typical definition, but it allows for easier immediate usage for the binomial case (where s=1).


ka'au
experimental cmavo mekso unary operator: cardinality (#,  ) Usually should be reserved for use on sets; if applied to group, it is the cardinality of the underlying set (Which is the order of the group) but it should probably not be applied to an element of a group. Application to a graph is ambiguous: is it the number of vertices or edges, or both, or neither, (if it defined at all)? For a set, each unique heretofore not counted element increments the running subtotal by 1 if the set is countable (small infinite or finite). See: cu'a, zilkancu, nilzilcmi, gu'au'i.


kei'ai
experimental cmavo mekso style converter: elementwise application of operator Prefixed to an operator/function that operates on numbers, thereby transforming it to a set operator (thus its arguments must be sets where before they were numbers), as defined in a given structure. Produces the set of all numbers that are given by some ordered pair of elements (the nth term of which belongs to the nth set specified) with the operator acting on them (per the rules of that operator). The set produced may include empty terms and/or infinity. Let "✦" represent the operator; then X_1 kei'ai ✦ X_2 boi X_3 boi X_4 = Set(x_{1}✦x_{2}✦x_{3}✦x_{4}: x_i in X_i). See also kei'au for a similar but different word.


la'e'au
experimental cmavo the specific referent of [following sumti] defined/specified by the grammar The grammar in question can itself be specified by metalinguistic tags that apply to the utterance; if not explicitly specified, then a cultural assumption/default is applied. This word is useful for distinguishing between, for example, differentiating between something that the speaker happens to call "PEMDAS" and the order of operations PEMDAS (which, presumably, could be included in the grammar as a special word that influences mekso interpretation); likewise for a font that happens to be called ".mekrot." ("math blackboard bold") and the one that definitely refers to the set of characters used for, among other purposes, set notation.


mau'au
experimental cmavo mekso: conversion of operator/function to operand Must be followed by a function (meaning "f", not "f(x)" (which is a number)) or operator (such as "+"). Treats it formally and syntactically as a number so that it can be operated upon, such as by se'au or by a functional/operator (technical sense). Terminated by zai'ai; the pair essentially act as mathematical quotation marks. Also works on connectives and kei'i. Expressions may be complicated (such as with SE, NA, NAI, etc. modifying the string); perhaps one may even find a use for quoting relations (such as equality or elementhood) and/or some brivla or other cmavo. See also: kei'au.


pau'a'u
experimental cmavo mekso operator: part of number/projection (one sense); the X2 part of X1 X1 can be a tensor, in which case the operator applies entrywise; X1 can be a function, in which case the operator applies pointwise. It extracts the part of the number that belongs to the structure X2 united with the singleton 0. X2 = R implies "real part"; X2 = i*R implies "imaginary part"; X2 = R+ implies "positive part"; X2 = R implies "negatige part". If X1 has no explicit value/projection in/along X2, then the output is 0. If 0 is X1 or X2, then the output is 0. X1 should really be a structure, not just a set.


xo'ei
experimental cmavo unary mekso operator: produces a string of n consecutive "xo'e"'s, treated as digits (concatenated into a single string of digits) n can be any nonnegative integer, infinity, or vague. The default value of n is itself "xo'e", thus transforming this word into a 0ary operator. The output is a string of digits, not a number; thus, it automatically attaches to a string of PA's if immediately preceding this word; following digits must be concatenated on; it might be necessary to convert this string into a number (especially if it is isolated and/or is being acted upon an operator that is defined as desired for numbers but not for digits or strings), but the conversion may be automatic according to the grammar (at least in some contexts). xo'e is considered to represent exactly one digit in the given base until it is converted into a number.


ju'au
experimental cmavo semimathematical binary operator: named number base operator/interpreter Follows a number and is followed by a sumti string (introduced with appropriate gadri, multiple sumti strung together via connectives); scope terminates with the end of the sumti and/or with ke'e'au. The construct as a whole syntactically functions as a parenthetical so that mekso may be resumed immediately thereafter without any difficulty. This word attaches to/affects the immediately preceding macrodigit string extending from the last evaluated number, number operation, etc. or bracket to/until this word. The preceding number is semantically interpreted according to the description to which the following sumti refers. If a sequence is supplied as the following sumti, the preceding number is interpreted so that each digit, starting from the left (default: see next), is understood as a multiple of the corresponding value in the sequence under the ordering given (which may produce an invalid/illformed result; the placement of pi matters as well). If a single number is submitted (such as "li dau"), this word acts as ju'u; the number is interpreted according to the rules of mekso and is its own island for such interpretation (thus ju'u is needed to override a cultural/grammatical default interpretation, such as decimal, even if this is done within the mekso expression outside of and including the scope of this word). In this way, the sumti being (10^n)_n, which is a sequence, is the same as the sumti being just 10, but is very different from it being the sequence (10)_n. The original default for numeralposition/string reading/interpretation is from lefttoright, where a digit on the left in a pair of digits represents that multiple of a "later" term in the sequence as compared to right member of the pair; thus "23" in decimal means (2*(10^1)) + (3*(10^0)), which is twentythree. This default may be overridden via specification of the sequence (either its ordering, its domain being negated, or alteration to the base); alternatively, and perhaps more easily, the appearance of "la'e zoi jbo. ri'u bi'o zu'a .jbo" will always reverse it (this is a special input value with interpretation defined by this grammar); when reversed, "23" in decimal means (2*(10^0)) + (3*(10^1)), which is to say thirtytwo. Generally, this word overrides the default interpretation of a string of microdigits when computing the value of a macrodigit (see: ju'u'i). See also: ju'u, ju'u'i, pi'e.


be'ei'oi
experimental cmavo ternary mekso operator: x_{1}th Bergelson multiplicative interval with exponents bounded from above by function x_{2} and with sequence of shifts x_{3}, where exponents belong to set x_{4} x1 must be a positive integer. x2 must be a strictly monotonic increasing function mapping from all of the positive integers to a subset (not necessarily proper) thereof. x3 must be a sequence of natural numbers. x2 without context will default to the same value as x1 (it is simple linear on the set of positive integers), x3 without context will be a sequence all and only of 1's, x4 without context defaults to the set of all nonnegative integers. Let p_i be a prime for all i, with p_{1} = 2 and the ith prime (in the normal monotonic increasing order) being p_i. Let all other symbols match the aforementioned conditions. Represent the nth term of the sequence x3 by x3_n; represent the function in x2 being applied to the number m by x2(m). Then x1 be'ei'oi x2 boi x3 boi x4 produces the set of all numbers of the form x3_(x1) * (p_{1})^(e_{1}) *...* (p_(x1))^(e_(x1)), where e_j belongs to the intersection of the interval [0, x2(x1)] with x4.


da'a'au
experimental cmavo mekso operatory: prime mark append For ordered inputs (A, B, C), where A is a general mathematical object, B is an integer (normally nonnegative), and C is a nonnegative integer; appends B prime(s)/distinguishing marks of type C to A. Presumably objects that have the same symbol as A but different B and C values are mutual(ly) related/analogs, but are at least formally distinguishable. B and C each default to 1. In practice, tick marks are usually called prime marks (and probably will effectively/culturally have a value of C=1); other distinguishing features are employed usually once these are used, such as overbars, overtildes/overtwiddles, dagger, asterisks, etc. For a fixed C, this operator forms a series in B for each A. The meaning is contextspecific. B = 0 iff C = 0; if A in unprimed (lacks any distinguishing marks), then B = 0; thus all mathematical objects have this operator implicitly attached to them with the value of 0 for the second and third argumentsize (note that this is not the default value for these arguments when this operator is explicitly mentioned). It is possible therefore to reference both primed and unprimed objects of form A simultaneously via reference to, for example, (1±1)/2 for the value for B and/or C (employing the setting of defaulting to the value of 1 as appropriate).


gu'au'i
experimental cmavo mekso operator, variable arity  algebraic structure order of X1; OR: order of/(size of) period of element X1 in algebraic structure X2 under operator/of type X3 If applied to an algebraic structure (such as a group) it gives the order thereof (which, for a group, is the cardinality of the underlying set). If applied to an element of an algebraic structure, one has the options to specify the structure in which its order is being considered and/or the operator with respect to which its order is being considered (for example, in a given ring, an elements additive order is usually not its multiplicative order), although either of these made remain vague and be inferred from context; order is the smallest nonnegative number of applications of the operator needed to be applied (in composition) to the original element in order for it to result in the identity element of the structure (thus, order is not always finite or even defined). See also: mau'au, cu'a.


kei'au
experimental cmavo mekso operator: finite result set derived from/on set A with/due to operator/function B under ordering of application C Equivalent to: lo'i li zy du ca'e li pe'o se'au mau'au B zai'ai vei ma'o xy boi ny ve'o boi tau sy boi C ku poi ke'a cmaci xanri zi'e poi ke'a mleca li ci'i zo'u tau sy klesi A. Acts on an operator/function (b) and produces all finite results of that operator being used on any allowable number of elements of the set A without repetition within any given application. The result must be defined (and finite, obviously). Application of the operator on nothing (the elements of the empty set) is generically allowed and follows convention (for example, an empty sum may evaluate to 0). Differs from kei'ai. Use mau'au and zai'ai for quoting B. C will be specified explicitly (possibly elsewhere) and/or via zoi'ai.


noi'a
experimental cmavo PA incidental relative clause; attaches to a PA number/numeral/digit with the ke'a referring to that PA number/numeral/digit. It attaches to the last mentioned PA, which can be only a single digit within a number rather than the entire number itself (if it has digit length greater than one in some base). Thus, beware of grouping. Confer: poi'a, jau'au. The attachment of this word to the immediately preceding word can be ovverriden by the usage of xe'au, which begins bracketing together an utterance/subexpression that will be considered to be the single and whole referent of the clause introduced by this word, wthe utterance of which also finishes the bracket. There is a proposal to expand the attachment of this word to any word spoken in a mekso utterance (rather than only members of selma'o PA), including members of selma'o VUhU, etc.


poi'a
experimental cmavo PA restrictive relative clause; attaches to a PA number/numeral/digit with the ke'a referring to that PA number/numeral/digit. It attaches to the last mentioned PA, which can be only a single digit within a number rather than the entire number itself (if it has digit length greater than one in some base). Thus, beware of grouping. Confer: noi'a, jau'au. The attachment of this word to the immediately preceding word can be ovverriden by the usage of xe'au, which begins bracketing together an utterance/subexpression that will be considered to be the single and whole referent of the clause introduced by this word, wthe utterance of which also finishes the bracket. There is a proposal to expand the attachment of this word to any word spoken in a mekso utterance (rather than only members of selma'o PA), including members of selma'o VUhU, etc.
