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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