## 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
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### References:

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