x_{1} is a binary operator which is associative in space/under conditions/on (or endowing) set x_{2}.

Denote x_{1} by "♤"; for any elements x,y,z in the set of concern, if '♤' is associative, then (x♤y)♤z = x♤(y♤z), where equality is defined in the space and parenthetic grouping denotes higher prioritization/earlier application. An operator being associative usually results, notationally, in the dropping of explicit parentheses. "Associative property"/"associativity" = "ka(m)( )socni". Notice that a space is understood to support association (or 'be associative') under a given operator x_{1} in the sense that its elements and equality relation are such that all ordered pairs can be exchanged in application under the operator while maintaining equality; this is similar to a space being endowed with identity element(s) under a given operator and also an operator have identity in a space - in English, associativity is often understood to be a property of a function and the existence of an identity to be a property of a space, but they are wed in Lojban, as they should be, for neither makes sense without the other and the property is a relationship between the operator and the space. See also: cajni, sezni, 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}. - 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.