# Serial relation

In set theory, a branch of mathematics, a serial relation, or a total relation, is a binary relation R for which every element of the domain has a corresponding range element (∀ xy  x R y).

For example, in ℕ = natural numbers, the "less than" relation (<) is serial. On its domain, a function is serial.

A reflexive relation is a serial relation but the converse is not true. However, a serial relation that is symmetric and transitive can be shown to be reflexive. In this case the relation is an equivalence relation.

If a strict order is serial, then it has no maximal element.

In Euclidean and affine geometry, the serial property of the relation of parallel lines $(m\parallel n)$ is expressed by Playfair's axiom.

In Principia Mathematica, Bertrand Russell and A. N. Whitehead refer to "relations which generate a series" as serial relations. Their notion differs from this article in that the relation may have a finite range.

For a relation R let {y: xRy } denote the "successor neighborhood" of x. A serial relation can be equivalently characterized as every element having a non-empty successor neighborhood. Similarly an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood". More commonly, an inverse serial relation is called a surjective relation, and is specified by a total converse relation.

In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.

## Algebraic characterization

Serial or total relations can be characterized algebraically by equalities and inequalities about relation compositions. If $R\subseteq X\times Y$ and $S\subseteq Y\times Z$ are two binary relations, then their composition R ; S is defined as the relation $R\ {;}\ S=\{(x,z)\in X\times Z\mid \exists y\in Y:(x,y)\in R\land (y,z)\in S\}.$ • Total relations R are characterized by the property that S ; R = ∅ implies S = ∅, for all sets W and relations SW×X, where ∅ denotes the empty relation.
• Let L be the universal relation: $\forall y\forall z.yLz$ . Another characterization of a total relation R is $R;L=L$ .
• A third algebraic characterization of a total relation involves complements of relations: For any relation S, if R is serial then ${\overline {R;S}}\subseteq R;{\bar {S}}$ , where ${\bar {S}}$ denotes the complement of $S$ . This characterization follows from the distribution of composition over union.:57
• A serial relation R stands in contrast to the empty relation ∅ in the sense that ${\overline {\emptyset ;L}}=L$ while ${\overline {R;L}}=\emptyset .$ :63

Other characterizations use the identity relation $I$ and the converse relation $R^{T}$ of $R$ :

• $I\subseteq R;R^{T}$ • ${\bar {R}}\subseteq R;{\bar {I}}.$ 