The Algebraic Theory of Semigroups, Volume II

Representations of Baer ∗-semigroups

“…These characterizations are formulated in terms of the existence of »-positive functions possessing certain properties. Such formulations have been used for general semigroups and »-semigroups [1], [10], [12], but the additional properties needed for Baer »-semigroups have not been previously considered.…”

Representations of semigroups and the translational hull of a regular Rees matrix semigroup

“…Introduction. The translational hull Q(S) of a weakly reductive semigroup S plays an important role in the construction of ideal extensions of 5 (4.4 of [3], [6]), and contains an isomorphic copy of S as a densely embedded ideal [5], [6]. The purpose of this paper is to show that a great number of representations of semigroups are essentially homomorphisms into the translational hull C1(S) of a suitable regular Rees matrix semigroup S=Jt°(G; I, A; P).…”

“…Notation and preliminaries. We generally follow the terminology used in [3] ; the notation is that introduced in [9] while undefined symbols have the meaning given to them in [3] (thus our notation sometimes differs from that in [3]). Multiplication is usually denoted by juxtaposition.…”

“…Let FD = ^°(ZZli;Z,A;F) where P is a A x Z matrix over ZZii with put =qur¡ if cj^ e Hxx, and 0 otherwise. Then F is a regular matrix and FD^Tr (D) ( [8,Theorem 6], see also p. 92 of [3]), so that TD is, up to an isomorphism, independent of the choice of Hxx, rh qu. For each i e I, fix an idempotent et e Rt and let r{ be the unique inverse of r¡ in Lx such that rt r{ = e¡ ; for every /x e A, fix an idempotent/, e F" and let q'u be the unique inverse of qu in Rx such that q'aqu=fi (note that r¡ri = e=quq¡1, see [3, (1) and (2) are simultaneously nonzero.…”

“…We have y.s^ßs where dßs={x e S | x^xí} and xßs = xs if x e dßs. It takes only little consideration to see that dßs = Ss~1, so that the assertion follows from Theorem 1.20 [3]. Alternatively, it follows from Theorem 6 that Y is a homomorphism, and from Theorem 6 and Proposition 2 that Y is one-to-one.…”