Quasi-Boolean powers of semilattices
This paper presents conditions for a groupoid to be embeddable into a quasi-Boolean power of a certain semilattice: Theorem. A groupoid \(G\) admits embedding into a quasi-Boolean power of a certain semilattice if and only if the following conditions hold: I. in \(G\) the identities \(xx=x\), \(xy=yx\), and \(x(xy)=xy\) are valid; II. on \(G\) there exists a congruence \(\theta\) all of whose classes are semilattices and the factor-groupoid \(G/\theta\) is also a semilattice; III. the relation \(\omega= \{(\alpha,\beta)\in G\times G\mid(\alpha,\beta)\in \theta\) and \(\alpha\beta= \alpha\}\) represents a stable order on \(G\).
scientific article; zbMATH DE number 1541504