Fayoumi, Hiba F. Locally-zero groupoids and the center of \(\text{Bin}(X)\). (English) Zbl 1216.20057 Commun. Korean Math. Soc. 26, No. 2, 163-168 (2011). 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. Cited in 4 Documents MSC: 20N02 Sets with a single binary operation (groupoids) 08A02 Relational systems, laws of composition Keywords:center; semigroups of binary systems; locally-zero groupoids PDFBibTeX XMLCite \textit{H. F. Fayoumi}, Commun. Korean Math. Soc. 26, No. 2, 163--168 (2011; Zbl 1216.20057) Full Text: DOI arXiv