# facni experimental gismu

x1 is an n-ary operator/map which is distributive/linear/homomorphic in or over or from space/structure x2, mapping to space or structure x3, thereby producing a new space/structure x4 which is the 'union' of x2 and x3 endowed with x1; x1 distributes over/through all of the operators of x2.

x2 and x3 cannot merely be sets; they must be structures which each are a set endowed with at least one operator each; the ith operator endowing one space corresponds to exactly the ith operator endowing the other space under mapping x1. For any operator of x2, x1 is commutative with it with respect to functional composition (fa'ai) when the (other) operator is 'translated' to the corresponding operator of x3 appropriately. x1 is linear/a linear operator; x1 is a homomorphism; x1 distributes. x2 is homomorphic with x3 under x1; they need not be identical (in fact, their respective operators need not even be identical, just 'homomorphically similar'). For "distributivity"/"distributive property", "linearity of operator", or "homomorphicity of operator", use "ka(m)( )facni" with x1 filled with "ce'u"; for "homomorphicity of spaces", use the same thing, but with x2 or x3 filled with "ce'u". See also: socni, cajni, sezni, dukni; fa'ai; fatri. This is a structure-operator-preserving function, and thus is an example of a stodraunju.

## In notes:

cajni
x1 is a binary operator which is commutative in space/under conditions/on (or endowing) set x2; x1 and x2 are each abelian (in different senses).
dukni
x1 is a binary operator in space/under conditions/on (or endowing) set x2 such that there is an identity element and for any element for which it is defined in the space (excepting possibly a small number of special elements), there exists at least one element also in the space which is a left-inverse of that element under the operator.
sezni
x1 is a binary operator such that there exists at least one (left-)identity element in space/under conditions/on (or endowing) set x2 under the operator.
socni
x1 is a binary operator which is associative in space/under conditions/on (or endowing) set x2.
stodraunju
x1 is a function mapping x2 (domain) to x3 (codomain) such that properties x4 (ka) of x2 are preserved in its image under x1 according to the rules/operations/relations of x3 corresponding to those of x2 by x1.