Arens regularity for totally ordered semigroups

Dales, H.G. and Strauss, D. (2022) Arens regularity for totally ordered semigroups. Semigroup Forum, 105 (1). 172–190. ISSN 0037-1912

Abstract

Let S be a semigroup. We shall consider the centres of the semigroup (βS,□) and of the algebra (M(βS),□), where M(βS) is the bidual of the semigroup algebra (ℓ1(S),⋆), and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of S ∗ and of M(S ∗) that are ‘determining for the left topological centre’ (DLTC sets) of βS and M(βS). It is known that, when the semigroup S is cancellative, ℓ1(S) is strongly Arens irregular and that there is a DLTC set consisting of two points of S ∗. In contrast, there is little that has been published about the Arens regularity of ℓ1(S) when S is not cancellative. Totally ordered, abelian semigroups, with the map (s, t) → s∧ t as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of βS and of M(βS) for all such semigroups, and give several examples, showing that the minimum cardinality of DTC sets may be arbitrarily large, and, in particular, we shall give an example of a countable, totally ordered, abelian semigroup S with this operation for which there is no countable DTC set for βS or for M(βS). There was no previously-known example of an abelian semigroup S for which βS or M(βS) did not have a finite DTC set.