composition of relations

Bisimulation can be defined in terms of composition of relations as follows.

Composition of relations will be associative if the factorization system is appropriately stable.

Unlike ordinary composition of relations, demonic composition is not associative.

In the presence of pullbacks and a proper factorization system, one can define the composition of relations.

From this definition, it's apparent one may define the joinability relation as , where is the composition of relations.

In Rel, composition of morphisms is exactly composition of relations as defined above.

Composition of relations is associative.

Just as there is composition of functions, there is composition of relations.

Note that, despite the notation, the converse relation is not an inverse in the sense of composition of relations: in general.

Other forms of composition of relations, which apply to general n-place relations instead of binary relations, are found in the join operation of relational algebra.

The set of binary relations over a set X, together with the composition of relations forms a monoid with zero, where the zero element is the empty relation (empty set).

Since all functions are binary relations, it is correct to use the fat semicolon for function composition as well (see the article on Composition of relations for further details on this notation).

In mathematics, demonic composition is an operation on binary relations which is somewhat comparable to ordinary composition of relations but is robust to refinement of the relations into (partial) functions or injective relations.

The objects of this allegory are sets, and a morphism X Y is a binary relation between X and Y. Composition of morphisms is composition of relations; intersection of morphisms is intersection of relations.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relation over X and Z: (see main article composition of relations)