×

zbMATH — the first resource for mathematics

Induced pseudoorders. (English) Zbl 0772.04001
Beside the well-known concepts of order and quasiorder, also the concept of pseudoorder is introduced; this is a reflexive and antisymmetric binary relation on a set. If \(R\) is a binary relation on a set \(A\), then \(\tau(R)=R\cup R^{-1}\), the symmetric closure of \(R\). If \(R\) is reflexive, then \(\tau(R)\) is a tolerance (a reflexive and symmetric relation) on \(A\). If \(R\) is a binary relation on \(A\), \(T\) is a tolerance on \(A\) and \(T\subseteq R\), then \(R/T\) is a binary relation on the set \(A/T\) of all blocks of \(T\) defined so that \(\langle B,C\rangle\in R/T\) for two blocks \(B\), \(C\) of \(T\) if and only if there exist elements \(b\in B\) and \(c\in C\) such that \(\langle a,b\rangle\in R\). A relation \(R\) is called weakly transitive, if \(\tau(R)\) is transitive; it is called semitransitive, if \(\langle a,b\rangle\in\tau(R)\), \(\langle b,c\rangle\in R\) imply \(\langle a,c\rangle\in R\), and \(\langle a,b\rangle\in\tau(R)\), \(\langle c,a\rangle\in R\) imply \(\langle c,b\rangle\in R\). A theorem on these concepts is proved; its corollary states that if \(R\) is a reflexive and semitransitive binary relation on \(A\neq \emptyset\), then \(\tau(R)\) is an equivalence on \(A\) and \(R/\tau(R)\) is a pseudoorder on \(A/\tau(R)\). Further the results are transferred from sets to lattices; compatible binary relations on lattices are studied.

MSC:
03E20 Other classical set theory (including functions, relations, and set algebra)
06A99 Ordered sets
06B99 Lattices
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] Chajda I.: Tolerance trivial algebras and varieties. Acta Sci. Math. (Szeged), 46 (1983), 35-40. · Zbl 0534.08001
[2] Chajda I.: Patritions, coverings and bocks of compatible relations. Glasnik Matem. (Zagreb), 14 (1979), 21-26. · Zbl 0402.08001
[3] Czédli G.: Factor lattices by tolerances. Acta Sci. Math. (Szeged), 44 (1982), 35-42. · Zbl 0484.06010
[4] Zelinka B.: Tolerance in algebraic structures. Czech. Math. J., 20 (1970), 240-256. · Zbl 0197.01002 · eudml:12529
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.