Homogeneous relation

In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X.[1][2][3] This is commonly phrased as "a relation on X"[4] or "a (binary) relation over X".[5][6] An example of a homogeneous relation is the relation of kinship, where the relation is between people.

Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric relation, and a general endorelation corresponding to a directed graph. An endorelation R corresponds to a logical matrix of 0s and 1s, where the expression xRy corresponds to an edge between x and y in the graph, and to a 1 in the square matrix of R. It is called an adjacency matrix in graph terminology.

  1. ^ Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN 978-1-4020-6164-6.
  2. ^ M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.
  3. ^ Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. Springer Science & Business Media. p. 496. ISBN 978-3-540-67995-0.
  4. ^ Mordeson, John N.; Nair, Premchand S. (8 November 2012). Fuzzy Mathematics: An Introduction for Engineers and Scientists. Physica. p. 2. ISBN 978-3-7908-1808-6.
  5. ^ Tanaev, V.; Gordon, W.; Shafransky, Yakov M. (6 December 2012). Scheduling Theory. Single-Stage Systems. Springer Science & Business Media. p. 41. ISBN 978-94-011-1190-4.
  6. ^ Meyer, Bertrand (29 June 2009). Touch of Class: Learning to Program Well with Objects and Contracts. Springer Science & Business Media. p. 509. ISBN 978-3-540-92145-5.