dukni experimental gismu

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.

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.

In notes:

klojyjoisocnyjoidukni
x1 is a binary group operator endowing set/space x2 ; x2 is the underlying set or the actual structure of a group with operator x1.
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).
facni
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.