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Generated preorders and equivalences. (English) Zbl 1012.08002
Summary: For any relation $$R$$, we denote by $$R^*$$ and $$R^\bullet$$ the smallest preorder and equivalence containing $$R$$, respectively. We establish some basic properties of the closures $$R^*$$ and $$R^\bullet$$. Moreover, we provide some new characterizations of equivalences in terms of generated preorders.
The results obtained naturally supplement some former statements of Árpad Száz on preorders and equivalences. Moreover, they can be applied to relators (relational systems). Namely, each topology can be derived from a preorder relator. Moreover, equivalence relators also frequently occur in the applications.

##### MSC:
 08A02 Relational systems, laws of composition
##### Keywords:
generated preorders; generated equivalences