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.

This presupposes sezni. This does not say that the operator has an inverse, merely that each element does under the operator (these properties are closely related though). See also: socni, sezni, cajni, facni.

- klojyjoisocnyjoidukni
- x
_{1}is a binary group operator endowing set/space x_{2}; x_{2}is the underlying set or the actual structure of a group with operator x_{1}. - cajni
- 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). - 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}.