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.

This does not say that the operator is the identity/trivial relation, merely that there is an identity element (these two properties are closely related though). See also: socni, cajni, dukni, 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). - 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}. - socni
- x
_{1}is a binary operator which is associative in space/under conditions/on (or endowing) set x_{2}.