Congruences on Bruck-Reilly extensions of monoids
In 1991, \textit{S. Rankin} started the study of congruences on Bruck-Reilly extensions of inverse monoids [Semigroup Forum 42, No. 3, 279-292 (1991; Zbl 0818.20076)]. In the paper under review the general case is investigated, that is, congruences on Bruck-Reilly extensions \(S=BR(T,\alpha)\) of arbitrary monoids \(T\). It turns out that there are only 3 types of congruences: (i) group-congruences, (ii) congruences \(\rho\) of the form: \((m,a,n)\rho(p,b,q)\) iff \(m=p\), \(n=q\), \((0,a,0)\rho(0,b,0)\), (iii) so-called cross-over congruences. First, all group-congruences on \(S=BR(T,\alpha)\) are described by means of those on \(S_0=BR(G,\alpha)\), where \(G\) denotes the group of units of \(T\). Since \(S_0\) is an inverse semigroup, all group-congruences are known by their kernels. Two alternative descriptions of congruences of type (i) on \(S=BR(T,\alpha)\) are given. Also, the least group-congruence on \(S\) is found to have the same form as that on inverse semigroups. Second, a construction of all cross-over congruences on \(S\) by means of ideals and congruences on \(T\) is given. In particular, it follows that no such congruence exists on \(S\) if \(T\) is simple. Finally, descriptions of the quotient semigroups of \(S\) by a congruence of type (ii) or (iii) as particular Bruck-Reilly semigroups are provided. Note: A summary of these results was published [in \textit{C. Bonzini et al.}, Conference on Semigroups, Luino/Italy 1992, 212-223 (1993; Zbl 0815.20055)].
scientific article; zbMATH DE number 736952
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