Binary relation

A term's definition may require additional properties that are not listed in this table.

belongs to the set of ordered pairs that defines the binary relation.

These include, among others: A function may be defined as a binary relation that meets additional constraints.

[6] A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.

This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.

, some authors define a binary relation or correspondence as an ordered triple

[citation needed] In a binary relation, the order of the elements is important; if

used here agrees with the standard notational order for composition of functions.

On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions.

Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment.

(A minor modification needs to be made to the concept of the ordered triple

, as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.

may be identified with a directed simple graph permitting loops, where

A strict partial order is a relation that is irreflexive, asymmetric, and transitive.

A total order is a relation that is reflexive, antisymmetric, transitive and connected.

[37] A strict total order is a relation that is irreflexive, asymmetric, transitive and connected.

may be subjected to closure operations like: Developments in algebraic logic have facilitated usage of binary relations.

Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of

The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices".

identity corresponds to difunctional, a generalization of equivalence relation on a set.

[39] Structural analysis of relations with concepts provides an approach for data mining.

This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block matrix with rectangular blocks of ones on the (asymmetric) main diagonal.

[44] In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management.

[47] The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones.

Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.

The notion of a general contact relation was introduced by Georg Aumann in 1970.

operator selects a boundary sub-relation described in terms of its logical matrix:

[53][54] The contrast of heterogeneous and homogeneous relations is highlighted by these definitions: There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other.

Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) between different sets