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Locally-zero groupoids and the center of \(\text{Bin}(X)\). (English) Zbl 1216.20057

Summary: We introduce the notion of the center \(Z\,\text{Bin}(X)\) in the semigroup \(\text{Bin}(X)\) of all binary systems on a set \(X\), and show that if \((X,\bullet)\in Z\,\text{Bin}(X)\), then \(x\neq y\) implies \(\{x,y\}=\{x\bullet y,y\bullet x\}\). Moreover, we show that a groupoid \((X,\bullet)\in Z\,\text{Bin}(X)\) if and only if it is a locally-zero groupoid.

MSC:

20N02 Sets with a single binary operation (groupoids)
08A02 Relational systems, laws of composition
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