Structure of concordant semigroups
The relations \({\mathcal L}^*\) and \({\mathcal R}^*\) in a semigroup S are defined as follows: elements \(a,b\in S\) are related by \({\mathcal L}^*\) (\({\mathcal R}^*)\) iff they are related by \({\mathcal L}\) (\({\mathcal R})\) in some oversemigroup of S. A semigroup is abundant if each \({\mathcal L}^*\)-class and each \({\mathcal R}^*\)-class contains an idempotent. Abundant semigroups are thus generalizations of regular semigroups. Parts of the theory of regular semigroups have been generalized to abundant semigroups in a series of papers by J. B. Fountain and his pupils. The present paper also belongs to this line. A concordant semigroup is an abundant semigroup in which the idempotents generate a regular subsemigroup, and which also satisfies a condition called idempotent-connectedness, relating the idempotents in the \({\mathcal L}^*\)-class of an element to those in the \({\mathcal R}^*\)-class of the element. E.g., full subsemigroups of regular semigroups or Rees matrix semigroups over a cancellative monoid with the sandwich matrix entries coming from the group of units, are concordant semigroups. The paper contains two main results. The first of them determines the structure of concordant semigroups in terms of inductive cancellative categories, and shows in fact that the categories of concordant semigroups and of inductive cancellative categories are equivalent. This generalizes the corresponding results of \textit{K. S. S. Nambooripad} [Mem. Am. Math. Soc. 224 (1979; Zbl 0457.20051)] on regular semigroups. (A cancellative category is a small category with left and right cancellation for morphisms; inductiveness is defined by conditions which in the groupoid case reduce to those used by Nambooripad.) The inductive cancellative category associated with a concordant semigroup S consists of, as morphisms, the triples \((e,x,f)\in S^ 3\), where e and f are idempotents belonging to the \({\mathcal R}^*\)-class and \({\mathcal L}^*\)- class, respectively, of x. - The second main result generalizes \textit{J. Meakin}'s description of regular semigroups [Proc. Edinb. Math. Soc., II. Ser. 21, 135-142 (1978; Zbl 0394.20046)] in terms of trace and structure mappings, anable functions. The question has been raised as to whether this restriction to twice differentiable functions is essential. In this paper we show that all \(C^ 1\) functions with derivative satisfying a Lipschitz condition can serve as policy functions and we also show by examples that differentiability is not a required property of a policy function.
category with left and right cancellation for morphisms
scientific article; zbMATH DE number 4065308
scientific article