# 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 (∀ *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 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 total 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 or total 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

- Total relations
*R*are characterized by the property that*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: . Another characterization of a total relation
*R*is .[7]

- A third algebraic characterization of a total relation involves complements of relations: For any relation
*S*, if*R*is serial then , where denotes the complement of . 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 while [5]^{:63}

Other characterizations use the identity relation and the converse relation of :

## References

- B. Russell & A. N. Whitehead (1910) Principia Mathematica, volume one, page 141 from University of Michigan Historical Mathematical Collection
- 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. - Gunther Schmidt (2011).
*Relational Mathematics*. Cambridge University Press. doi:10.1017/CBO9780511778810. ISBN 9780511778810. Definition 5.8, page 57. - 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 - 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. - If
*S*≠ ∅ and*R*is total, then implies , hence , hence . The property follows by contraposition. - Since
*R*is serial, the formula in the set comprehension for*P*is true for each*x*and*z*, so . - If
*R*is serial, then , hence .

- 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..