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.


03E20 Other classical set theory (including functions, relations, and set algebra)
06A99 Ordered sets
06B99 Lattices
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