x_{1} is a binary operator which is commutative in space/under conditions/on (or endowing) set x_{2}; x_{1} and x_{2} are each abelian (in different senses).

Denote x_{1} 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.

- dukni
- x
_{1}is a binary operator in space/under conditions/on (or endowing) set x_{2}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. - facni
- x
_{1}is an n-ary operator/map which is distributive/linear/homomorphic in or over or from space/structure x_{2}, mapping to space or structure x_{3}, thereby producing a new space/structure x_{4}which is the 'union' of x_{2}and x_{3}endowed with x_{1}; x_{1}distributes over/through all of the operators of x_{2}. - sezni
- x
_{1}is a binary operator such that there exists at least one (left-)identity element in space/under conditions/on (or endowing) set x_{2}under the operator. - socni
- x
_{1}is a binary operator which is associative in space/under conditions/on (or endowing) set x_{2}.