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Zbl 1012.08003
Glavosits, Tamás
Preorders and equivalences generated by commuting relations. (English)
[J] Acta Math. Acad. Paedagog. Nyházi. (N.S.) 18, 53-56 (2002). ISSN 1786-0091/e

Two binary relations $R$, $S$ on a set $X$ are called commuting, if $R\circ S=S\circ R$, where the symbol $\circ$ denotes the composition of relations. The paper starts by the study of compositions of powers of commuting relations $R$, $S$. The main result states that $(R\cup S)^n=\bigcup^n_{k=0}S^{n-k}\circ R^k$ holds for them. For every binary relation $R$ on $X$ the relation $R^*=\bigcup^\infty_{n=1}R^n$ is defined; this is the smallest (with respect to set inclusion) preorder on $X$ containing $R$. Further, the smallest equivalence $(R\cup R^{-1})^*$ on $X$ containing $R$ is defined. It is also denoted by $R$ with asterisk, but that asterisk is bigger. The properties of these concepts are studied.
[B.Zelinka (Liberec)]

MSC 2000:: *08A02 General relational systems

Keywords: generated equivalences; commutativity of composition; commuting relation; power of a relation; preorder relation; equivalence relation; generated preorders

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