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 (∀ x ∃ y  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 ${\displaystyle (m\parallel n)}$ is expressed by Playfair's axiom.

In Principia Mathematica, Bertrand Russell and A. N. Whitehead refer to "relations which generate a series"[1] 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".[2] More commonly, an inverse serial relation is called a surjective relation, and is specified by a serial converse relation.[3]

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

Algebraic characterization

Serial relations can be characterized algebraically by equalities and inequalities about relation compositions. If ${\displaystyle R\subseteq X\times Y}$ and ${\displaystyle S\subseteq Y\times Z}$ are two binary relations, then their composition R ; S is defined as the relation ${\displaystyle R\ {;}\ S=\{(x,z)\in X\times Z\mid \exists y\in Y:(x,y)\in R\land (y,z)\in S\}.}$

• If R is a serial relation, then S ; R = ∅ implies S = ∅, for all sets W and relations S ⊆ W×X, where ∅ denotes the empty relation.[5][6]
• Let L be the universal relation: ${\displaystyle \forall y\forall z.yLz}$. A characterization of a serial relation R is ${\displaystyle R;L=L}$.[7]
• Another algebraic characterization of a serial relation involves complements of relations: For any relation S, if R is serial then ${\displaystyle {\overline {R;S}}\subseteq R;{\bar {S}}}$, where ${\displaystyle {\bar {S}}}$ denotes the complement of ${\displaystyle S}$. This characterization follows from the distribution of composition over union.[5]^:57[8]
• A serial relation R stands in contrast to the empty relation ∅ in the sense that ${\displaystyle {\overline {\emptyset ;L}}=L}$ while ${\displaystyle {\overline {R;L}}=\emptyset .}$[5]^:63

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

• ${\displaystyle I\subseteq R;R^{T}}$
• ${\displaystyle {\bar {R}}\subseteq R;{\bar {I}}.}$[5][3]

References

1. B. Russell & A. N. Whitehead (1910) Principia Mathematica, volume one, page 141 from University of Michigan Historical Mathematical Collection
2. Yao, Y. (2004). "Semantics of Fuzzy Sets in Rough Set Theory". Transactions on Rough Sets II. Lecture Notes in Computer Science. 3135. p. 309. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.
3. Gunther Schmidt (2011). Relational Mathematics. Cambridge University Press. doi:10.1017/CBO9780511778810. ISBN 9780511778810. Definition 5.8, page 57.
4. James Garson (2013) Modal Logic for Philosophers, chapter 11: Relationships between modal logics, figure 11.1 page 220, Cambridge University Press doi:10.1017/CBO97811393421117.014
5. Schmidt, Gunther; Ströhlein, Thomas (6 December 2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. p. 54. ISBN 978-3-642-77968-8.
6. If S ≠ ∅ and R is serial, then ${\displaystyle \exists w\exists x.wSx}$ implies ${\displaystyle \exists w\exists x\exists y.wSx\land xRy}$, hence ${\displaystyle \exists w\exists y.w(S;R)y}$, hence ${\displaystyle S;R\neq \emptyset }$. The property follows by contraposition.
7. ${\displaystyle P=R;L=\{(x,z):\exists y.xRy\land yLz\}}$ Since R is serial, the formula in the set comprehension for P is true for each x and z, so ${\displaystyle P=L}$.
8. If R is serial, then ${\displaystyle L=R;L=R;(S\cup {\bar {S}})=(R;S)\cup (R;{\bar {S}})}$, hence ${\displaystyle {\overline {R;S}}\subseteq R;{\bar {S}}}$.

• Jing Tao Yao and Davide Ciucci and Yan Zhang (2015). "Generalized Rough Sets". In Janusz Kacprzyk and Witold Pedrycz (ed.). Handbook of Computational Intelligence. Springer. pp. 413–424. ISBN 9783662435052. Here: page 416.
• Yao, Y.Y.; Wong, S.K.M. (1995). "Generalization of rough sets using relationships between attribute values" (PDF). Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..