Orthodox semigroups whose idempotents satisfy a certain identity

The algebraic theory of inverse semigroups has received considerable attention in recent years with satisfactory structure theorems being obtained for extensive classes of these semigroups. As is the case with so much of semigroup theory, many of these structure theorems were obtained by, or had their origin in fundamental papers of, A. H. Clifford. For example, the structure of inverse semigroups with central idempotents (semilattices of groups) was obtained by Clifford in [1], that of Brandt semigroups was obtained in [2], while, in [3], Clifford described bisimple inverse semigroups in terms of their right unit subsemigroup. The third of these papers has been the basis for much of the research on structural aspects of inverse semigroups. On the one hand, its direct descendants include Reilly's papers on bi-simple inverse semigroups in terms of RP-systems [10], [11], and the author's description of O-bisimple inverse semigroups in terms of groups and semilattices [5]. On the other hand, it has led to Reilly's description [9] of w-bisimple inverse semigroups and Warne's characterisations of various classes of bisimple inverse semigroups with special semilattices of idempotents. These, in turn, have led to the determination of simple inverse semigroups with special semilattices of idempotents.

We investigate recent uniqueness theorems for reduced \(C^*\)-algebras of Hausdorff etale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full \(C^*\)-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.

A RANDOM RAMBLE THROUGH INVERSE SEMIGROUPS

Using the classification and description of the structure of bisimple monogenic orthodox semigroups obtained in \cite{key10}, we prove that every bisimple orthodox semigroup generated by a pair of mutually inverse elements of infinite order is strongly determined by the lattice of its subsemigroups in the class of all semigroups. This theorem substantially extends an earlier result of \cite{key25} stating that the bicyclic semigroup is strongly lattice determined.

SynopsisA regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.

A construction of all coextensions of a regular semigroup $S$ by rectangular bands is obtained. The construction is analogous to Hall’s construction of orthodox semigroups as spined products of bands and inverse semigroups and reduces to that construction when $S$ is inverse. The results are specialized to provide a construction of the category of all normal coextensions of a regular semigroup.

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band ES of idempotents of S such that F∋e, ES∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of FS.

A semigroup S is called E-inversive if for every a ∈ S ther is an x ∈ S such that (ax)2 = ax. A construction of all E-inversive subdirect products of two E-inversive semigroups is given using the concept of subhomomorphism introduced by McAlister and Reilly for inverse semigroups. As an application, E-unitary covers for an E-inversive semigroup are found, in particular for those whose maximum group homomorphic image is a given group. For this purpose, the explicit form of the least group congruence on an arbitrary E-inversive semigroup is given. The special case of full subdirect products of a semilattice and a group (that is, containing all indempotents of the direct product) is investigated and, following an idea of Petrich, a construction of all these semigroups is provided. Finally, all periodic semigroups which are subdirect products of a semilattice or a band with a group are characterized.

In [1] Clifford showed that the structure of any bisimple inverse semigroup with identity is uniquely determined by that of its right unit subsemigroup. The object of this paper is to show that the structure of any bisimple inverse semigroup with or without identity is determined by that of any of its a-classes. Let us define a right partial semigroup S to be a set S together with a partial binary operation satisfying the following condition: (A) if, for elements a, b, c of S, a(bc) is defined then so also is (ab)c defined and then a(bc) = (ab)c. We say that a right partial semigroup S is isomorphic with a right partial semigroup T if there exists a bijection b of S onto T such that ab is defined if and only if aq bq is defined and such that if ab is defined then (ab)k = aq bb. We define an RP-system (R, P) to be a right partial semigroup R together with a subsemigroup P of R such that: P(1) ab is defined if and only if a E P, for all a, b in R; P(2) R has a left identity contained in P; P(3) ac = bc implies that a = b for all a, b E P, c E R; P(4) foralla,be R,PanPb=Pcforsomece R. It then follows from P(1) and P(3) that any left identity of R contained in P is, in fact, a two-sided identity for P and so is unique. Now consider any a-class R of a bisimple inverse semigroup S. If we define the partial binary operation o in R by the rule that a o b = ab if and only if ab e R, then with respect to this operation R is a right partial semigroup with a subsemigroup P such that (R, P) is an RP-system (Theorem 1.4). Note 1. RP-systems can be obtained from systems having products more generally defined than stipulated in P(1). We then just ignore products which do not satisfy the condition P(1) (cf. Examples 1, 2 in ?6). Note 2. In particular, R could be a lattice ordered group and P the positive cone of R. We show that for any RP-system (R, P) there exists a bisimple inverse semigroup, which we denote by R-1 o R, some a-class of which is isomorphic with R. Conversely, for any a-class R of a bisimple inverse semigroup S there is a subsemigroup P of R such that (R, P) is an RP-system and R1 o R is isomorphic with S.

The structure of a bisimple inverse semigroup with an identity has been related by Clifford [2] to that of its right unit subsemigroup. In this paper we give an explicit structure theorem for bisimple inverse semigroups in which the idempotents form a simple descending chaine0 > e1 > e2.…We call such a semigroup a bisimple co-semigroup. The structure of a semigroup of this kind is shown to be determined entirely by its group of units and an endomorphism of its group of units.

For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.

The structure of E-unitary inverse semigroups has been described by McAlister and by Reilly and the author. The parameters in the first structure theorem may be made into a category, and the same holds for the parameters in the second structure theorem. We prove that each of these categories is equivalent to the category of E-unitary inverse semigroups and their homomorphisms. We also provide functors between the two first-mentioned categories which are naturally equivalent to the composition of the functors figuring in the categorical equivalence referred to above.

We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.

Let [Formula: see text] be the semigroup of all singular order-preserving mappings on the finite set [Formula: see text]. It is known that [Formula: see text] is generated by its set of idempotents of rank [Formula: see text], and its rank and idempotent rank are [Formula: see text] and [Formula: see text], respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank [Formula: see text] in [Formula: see text]. We also calculate its rank and idempotent rank.

In this paper we define the module topological center of the second dual \(\mathcal{A}^{**}\) of a Banach algebra \(\mathcal{A}\) which is a Banach \(\mathfrak{A}\)-module with compatible actions on another Banach algebra \(\mathfrak{A}\). We calculate the module topological center of ℓ 1(S)**, as an ℓ 1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ℓ 1(S)** is ℓ 1(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.