The synthesis theorem for finite regular semigroups, and its generalization (Q1110661)

The paper is a detailed account of the results announced by the author in the Proceedings of the 1984 Marquette Conference on Semigroups [1-15 (1985; Zbl 0564.20036)]. The aim is to improve on the previous research of Allen and Rhodes which showed that any finite semigroup divides a finite iterative structure-matrix semigroup. The author's modified approach is not inductive and applies to a wide class of infinite semigroups. Extensions of matrix semigroups by 0-disjoint unions of groups (not just a single group) are permitted and the group elements act as partial permutations on the base sets of the matrix semigroups. An unambiguous semigroup S is one in which the posets S/\({\mathcal R}\), S/\({\mathcal L}\) have unambiguous orders (x\(\leq z\) and \(y\leq z\) \(\Rightarrow\) x and y are comparable). Any semigroup has an unambiguous expansion which inherits the properties of finiteness and regularity. S is unambiguous- except-at-0 if either S is unambiguous, or S has a zero 0 and the orders on S/\({\mathcal R}\) and S/\({\mathcal L}\) are unambiguous when restricted to S- \(\{\) \(0\}\). S is stable iff \(R_ x\leq R_ y\) or \(L_ x\leq L_ y\) \(\Rightarrow\) \(J_ x\leq J_ y\). The synthesis theorem for regular semigroups then states that if S is regular, unambiguous-except-at-0, stable and there is a bound on the length of ascending J-chains from each J-class of S then \(S\leq U\leq V\leq IMS\) where the semigroup \(U=Reg(V)\) is regular, IMS is an iterative structure-matrix semigroup and (IMS-V)\(\cup \{0\}\) is a nilpotent ideal of index 3. Moreover \(S^ 0\) is a retract of IMS. If S is finite, S and IMS share the same complexity. The synthesis theorem for the nonregular case is similar and is based on the fact that a non-regular semigroup S as above can be embedded in a regular semigroup \((S)_{reg}\) with the same properties.