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In set theory, a serial relation is a binary relation R for which every element of the domain has a corresponding range element (∀ xy  x R y). Serial relations are sometimes called total relations, but the term total relation has also been used to designate a connex relation, i.e., a homogeneous binary relation R for which either xRy or yRx holds for any pair (x,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]

In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.[3]

Algebraic characterizationEdit

Serial relations can be characterized algebraically by equalities and inequalities about relation compositions. If   and   are two binary relations, then their composition   is defined as the relation  

  • Total relations R are characterized by[clarify] the property that RS = ∅ implies S = ∅, for all sets W and relations SW×X, where ∅ denotes the empty relation.[4][5]
  • Let L be the universal relation:  . Another characterization[clarify] of a total relation R is  .[6]
  • A third algebraic characterization[clarify] 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.[4]:57[7]
  • A serial relation R stands in contrast to the empty relation ∅ in the sense that   while  [4]:63

Other characterizations[clarify] use the identity relation   and the converse relation   of  :

  •  
  •  [4]

ReferencesEdit

  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. ^ 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
  4. ^ a b c d 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.
  5. ^ If S ≠ ∅ and R is total, then   implies  , hence  , hence  . The property follows by contraposition.
  6. ^   Since R is serial, the formula in the set comprehension for P is true for each x and z, so  .
  7. ^ If R is serial, then  , hence  .