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# Serial relation

<https://en.wikipedia.org/wiki/Serial_relation> <https://proofwiki.org/wiki/Definition:Left-Total_Relation> <https://proofwiki.org/wiki/Definition:Right-Total_Relation> <https://proofwiki.org/wiki/Inverse_of_Left-Total_Relation_is_Right-Total>

**A serial or left-total relation** is a binary relation R for which every element of the domain has a corresponding range element, `∀x∃y.xRy`.

Examples:

* a function is serial on its domain
* "less than" relation on ℕ 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 ∥ 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 serial converse relation.

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