cajni experimental gismu

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).

Denote x1 by "♤"; for any elements x,y in the set of concern, if '♤' is commutative, then x♤y = y♤x, where equality is defined in the space. "Commutative property"/"commutativity" = "ka(m)( )cajni". See also: socni, sezni, dukni, facni.

In notes:

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.
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.