Semilattice decompositions of semigroups revisited
Let \(N\) denote the set of all positive integers. Let \(S\) be a semigroup, and \(a,b\in S\). Define relations \(\mid\), \(\to\), and \(-\), respectively by \(a\mid b\Leftrightarrow (\exists x,y\in S^1) b=xay\), \(a\to b\Leftrightarrow (\exists n\in N) a\mid b^n\), \(a-b\Leftrightarrow a\to b\to a\). We associate a digraph \((S,\to)\) and a graph \((S,-)\) corresponding to \(\to\) and \(-\), respectively. If there exists a path from \(a\) to \(b\) in \((S,\to)\) (respectively, a path between \(a\) and \(b\) in \((S,-)\)) then there exist paths from \(a\) to \(b\) (respectively between \(a\) and \(b\)) of minimal length. Those are called the minimal paths from \(a\) to \(b\) (respectively between \(a\) and \(b\)). The rank of \(S\), \(\text{ran}(S)\), is the supremum of the lengths of all minimal paths in \((S,-)\), and the semirank of \(S\), \(\text{sran}(S)\), is the supremum of the lengths of all minimal paths in \((S,\to)\). The \(n\)-th powers of \(\to\) and \(-\) are, respectively, denoted by \(\to^n\) and \(-^n\). Then \(\text{ran}(S)\) is the smallest \(n\leq\infty\) for which \(-^n\) is transitive, and \(\text{sran}(S)\) is the smallest \(n\leq\infty\) for which \(\rightarrow^n\) is transitive. Next for \(n\in N\) and \(a\in S\) define the subsets \(\Sigma_n(a)\) and \(\Sigma(a)\) of \(S\) as follows: \(\Sigma_n(a)=\{x\in S\mid a\to^n x\}\), \(\Sigma(a)=\{x\in S\mid a\to^\infty x\}\) and define \(\sigma_n\) and \(\sigma\) on \(S\): \((a,b)\in\sigma_n\Leftrightarrow\Sigma_n(a)=\Sigma_n(b)\), \((a,b)\in\sigma\Leftrightarrow\Sigma(a)=\Sigma(b)\). Further, let \(\widehat\Sigma_n(a)=\{x\in S\mid a-^nx\}\), \((a,b)\in\widehat\sigma_n\Leftrightarrow\widehat\Sigma_n(a)=\widehat\Sigma_n(b)\). According to Ćirić and Bogdanović, \(\sigma\) is the smallest semilattice congruence on \(S\). Then \(S\) is \(\sigma_n\)-simple if and only if \(a-^nb\) for all \(a,b\in S\). Also \(\widehat\sigma_n\)-simpleness implies \(\sigma_n\)-simpleness. In this paper the structure of \(S\) belonging to \(\widehat{\mathcal S}_n\) is characterized in many ways, that is, the following are equivalent. (1) \(-^n\) is transitive, (2) \(-^n\) is a semilattice congruence on \(S\), (3) \(\widehat\sigma_n\) is a band congruence on \(S\), (4) \(S\) is a semilattice of \(\widehat\sigma_n\)-simple semigroups, (5) \(-^n=\sigma_n\), also four other equivalent conditions. As an application of the above result, the authors characterize a semilattice of Archimedean semigroups in terms of the relation \(-\). Next the left hand analogue is considered. Define the following. \(\Lambda_n(a)=\{x\in S\mid a\to^n x\}\), \(\Lambda(a)=\{x\in S\mid a\to^\infty x\}\), \((a,b)\in\lambda_n\Leftrightarrow \Lambda_n(a)=\Lambda_n(b)\), \((a,b)\in\lambda\Leftrightarrow\Lambda(a)=\Lambda(b)\). \(\widehat\Lambda_n(a)=\{x\in S\mid a-^nx\}\), \(\widehat\Lambda(a)=\{x\in S\mid a-^\infty x\}\), \((a,b)\in\widehat\lambda_n\Leftrightarrow\Lambda_n(a)=\Lambda_n(b)\), \((a,b)\in\widehat\lambda\Leftrightarrow\widehat\Lambda(a)=\widehat\Lambda(b)\). A semilattice of \(\widehat\lambda\)-simple semigroups, a semilattice of \(\widehat\lambda_n\)-simple semigroups and a semilattice of left Archimedean semigroups are characterized in terms of \(\overset{\ell^n} -\) or \(\overset\ell -\) etc. Finally, the radical of Green's \(\mathcal J\)-relation is introduced: For any relation \(\rho\) \((a,b)\in R(\rho)\Leftrightarrow(\exists m,n\in N) a^m\rho b^n\), \((a,b)\in T(\rho)\Leftrightarrow (\exists n\in N) a^n\rho b^n\). The authors discuss when \(R({\mathcal J}^n)\), \(R({\mathcal J})\), \(T({\mathcal J})^n\) and \(T({\mathcal J})\) are semilattices congruences. This paper gives refinements and further developments of results of the reviewer, M. S. Putcha, L. N. Shevrin and the first two authors of this paper.
scientific article; zbMATH DE number 1523492