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.


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.
