Rees matrix covers for regular semigroups
A regular semigroup S is called locally P (for some property P) if each local submonoid eSe has P (where \(e\in E(S))\). A homomorphism from a regular semigroup is called a local isomorphism if it is one-to-one on all local submonoids. In [Trans. Am. Math. Soc. 277, 727-738 (1983; Zbl 0516.20039)] the author showed that S is locally inverse if and only if S is a locally isomorphic image of a regular Rees matrix semigroup over an inverse semigroup. In this paper he proves analogs of this theorem for locally E-solid semigroups (regular semigroups whose idempotent generated part is completely regular), for locally orthodox semigroups and for locally L-unipotent semigroups. He proves that S is locally E-solid [locally orthodox, locally L-unipotent] if and only if S is a locally isomorphic image of a regular Rees matrix semigroup over an E-solid [orthodox, L-unipotent] semigroup. The proof utilises a construction which begins with a Rees matrix semigroup \({\mathcal M}(S;E,E;Q)\), where \(E=E(S)\) and Q is any \(E\times E\) matrix such that (i) \(q_{f,e}\in fSe\) and (ii) \(q_{e,e}=e\), for all e,\(f\in E\). One then considers the regular subsemigroup \(W=\{(e,x,f):\quad x\in eSf\}.\) The key result is that if \(\psi\) :\(W\to T\) is a local isomorphism onto a regular semigroup T, then S is a locally isomorphic image of a regular Rees matrix semigroup over T. This is coupled with results of \textit{T. E. Hall} [Semigroup Forum 25, 35-48 (1982; Zbl 0497.20049)] to obtain these covering theorems. Locally isomorphic images of regular Rees matrix semigroups over completely regular semigroups are characterized as those regular semigroups which are locally complete regular and whose principal ideals form a semilattice under intersection. Finally it is shown that if a locally E-solid semigroup S is the multiplicative semigroup of a ring, then S is in fact a semilattice of groups.
scientific article; zbMATH DE number 3863480
scientific article