Serial relation

In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.

Bertrand Russell used serial relations in The Principles of Mathematics^[1] (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.^[2]

A serial relation R is an endorelation on a set U. As stated by Russell, ${\displaystyle \forall x\exists y\ xRy,}$ where the universal and existential quantifiers refer to U. In contemporary language of relations, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest.

For a relation R, let {y: xRy} denote the "successor neighborhood" of x. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood".^[3]

In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.^[4]