A new approach to some greatest decompositions of semigroups (Q1345728)

This paper is to report briefly the new results of the authors with respect to greatest decompositions of semigroups with zero, that is, two topics: 1. semilattice decompositions and 2. decompositions of semigroups. The two kinds of relations play an important part. On a semigroup \(S\), define a relation \(\to\) by \(a\to b\) iff \((\exists n\in Z^+)\;b^n\in J(a)\), \(a,b\in S\), where \(J(a)\) is the \(\mathcal J\)-class containing \(a\). For each \(a\in S\), \(n\in Z^+\), define \(\Sigma_n (a) =\{x\in S : a\to^nx\}\), \(\Sigma(a) =\{x\in S : a\to^\infty x\}\), \(a\sigma b\) iff \(\Sigma(a) =\Sigma (b)\), \(a\sigma_n b\) iff \(\Sigma_n(a) =\Sigma(b)\). Next, \(a\to^l b\) iff \((\exists n\in Z^+)\;b^n\in L(a)\) where \(L(a)\) is the \(\mathcal L\)-class containing \(a\). Also \(\Lambda_n(a) =\{x\in S : a\to^{l^n}x\}\), \(\Lambda(a) =\{ x\in S : a\to^{l^\infty}x\}\), \(a\lambda b\) iff \(\Lambda(a) =\Lambda(b)\), \(a\lambda^n b\) iff \(\Lambda_n(a) =\Lambda_n(b)\). The following are presented for the first topic: (1) The greatest semilattice homomorphic image of a semigroup is characterized in terms of principal radicals. (2) A generalization of archimedeaness in terms of the relations from archimedeaness to semilattice-indecomposability (i.e. \(\sigma_1\)-simplicity, \(\cdots\), \(\sigma_n\)-simplicity, \(\cdots\), \(\sigma\) \((=\sigma_\infty)\)-simplicity). (3) A characterization of semigroups whose greatest semilattice homomorphic image is a chain (a chain on \(\sigma\)-, \(\sigma_n\)-simple semigroups etc.). (4) A characterization of semilattices of \(\lambda\)- or \(X_n\)-simple semigroups or semilattices of left archimedean semigroups. Decomposition of semigroups with zero needs a different idea from the first topic. Two methods, orthogonal decompositions and decompositions into a right sum of semigroups are employed. Let \(S\) be a semigroup with zero, and assume \(S =\bigcup_{\alpha\in Y} S_\alpha\) where \(S_\alpha\neq 0\) for all \(a\in Y\), \(S_\alpha\cap S_\beta = S_\alpha S_\beta =\{0\}\) for all \(\alpha,\beta\in Y\), \(\alpha\neq\beta\). Then the family \(\{S_\alpha :\alpha\in Y\}\) is called an orthogonal decomposition of \(S\) and the \(S_\alpha\) are called orthogonal summands of \(S\). A semigroup \(S\) with 0 is a right sum of semigroups \(S_\alpha\neq 0\), \(\alpha\in Y\) if \(S =\bigcup_{\alpha\in Y} S_\alpha\) such that \(S_\alpha\cap S_\beta = 0\) and \(S_\alpha S_\beta\subseteq S_\beta\) for all \(\alpha,\beta\in Y\), \(\alpha\neq\beta\). Define two kinds of relations on \(S\). \(x\sim y\) iff \(J(x)\cap J(y)\neq 0\) for \(x, y\in S^*\), \(0\sim 0\). Next, \(\Delta_n(a) =\{x\in S: x\sim^n a\}\cup\{0\}\), \(\Delta(a) =\{x\in S : x\sim^\infty a\}\cup\{0\}\), \(a\delta b\) iff \(\Delta (a) =\Delta(b)\), \(a \delta_n b\) iff \(\Delta_n(a) = \Delta(b)\). Also \(x\sim^l y\) iff \(L(x) \cap L(y) \neq 0\) for all \(x, y \in S^*\) \((= S\setminus \{0\})\), \(0\sim^l 0\). \(K_n(a) = \{x \in S : x\sim^{l^n} a \} \cup \{0\}\), \(a \kappa b\) iff \(K(a) = K(b)\), \(a \kappa_n b\) iff \(K_n(a) = K_n(b)\) (\(\sim^r\) is dually defined). The following results are obtained: (1) Every semigroup with zero has a greatest orthogonal decomposition and every summand of this decomposition is an orthogonal indecomposable semigroup. (2) Every semigroup with 0 has a greatest decomposition into a right sum. (3) A characterization of orthogonal sums of \(0\)-\(\delta_n\)-simple (\(0\)-\(\sigma\)-simple, \(0\)- \(\sigma_n\)-simple) semigroups. Finally the authors present connections between orthogonal decompositions (decompositions into a right sum) of a semigroup with \(0\) and decompositions of the lattice of its ideals (left ideals) into a direct product. The reviewer appreciates the authors' complete study of the situation of semilattice components between archimedeaness and semilattice indecomposability. The second topic has been much developed by the authors. This paper is a survey including the authors' result and comprehensive references.