Serial relation


In set theory, a branch of mathematics, a serial relation, also called a left-total relation, is a binary relation R for which every element of the domain has a corresponding range element.
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 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 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 serial 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 relations can be characterized algebraically by equalities and inequalities about relation compositions. If and are two binary relations, then their composition R ; S is defined as the relation
Other use the identity relation and the converse relation of :
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