Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 18, No. 2, pp. 53-56 (2002)

Preorders and equivalences generated by commuting relations

T. Glavosits

University of Debrecen

Abstract: For any relation $R$ on a fixed set $X$, we denote by $R^{\star}$ and $R^{\bigstar}$ the smallest preorder and equivalence on $X$ containing $R$, respectively.

We show that if $R$ and $S$ are commuting relations on $X$ in the sense that $R\circ S=S\circ R$, then $R^{\star}\circ S=S\circ R^{\star}$, $R\circ S^{\star}=S^{\star}\circ R$ and $R^{\star}\circ S^{\star}=S^{\star}\circ R^{\star}$.

Moreover, we show if in addition to the condition $R\circ S=S\circ R$ we also have $R\circ S^{-1}\!=S^{-1}\!\circ R$, then the corresponding equalities hold for the operation $\bigstar$ too.

Keywords: Preorders and equivalences, commutativity of composition.

Classification (MSC2000): 08A02

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