Victoria Gould

THE FREE AMPLE MONOID

We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice of idempotents of FIM(X) and that FAM(X) is embedded in . Finally we show that every weakly E-ample monoid has a proper ample cover.

Restriction Semigroups and Inductive Constellations

The Ehresmann–Schein–Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample, and weakly E-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals. Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial actions and expansions for inductive constellations, along the lines of that of Gilbert for inductive groupoids.

Graph expansions of unipotent monoids

Margolis and Meakin use the Cayley graph of a group presentation to construct E-unitary inverse monoids [11]. This is the technique we refer to as graph expansion. In this paper we consider graph expansions of unipotent monoids, where a monoid is unipotent if it contains a unique idempotent. The monoids arising in this way are E-unitary and belong to the quasivariety of weakly left ample monoids. We give a number of examples of such monoids. We show that the least unipotent congruence on a weakly left ample monoid is given by the same formula as that for the least group congruence on an inverse monoid and we investigate the notion of proper for weakly left ample monoids. Using graph expansions we construct a functor Fe from the category U of unipotent monoids to the category PWLA of proper weakly left ample monoids. The functor Fe is an expansion in the sense of Birget and Rhodes [2]. If we equip proper weakly left ample monoids with an extra unary operation and denote the corresponding category by PWLA 0 then regarded as a functor U→PWLA 0 Fe is a left adjoint of the functor Fσ : PWLA 0 → U that takes a proper weakly left ample monoid to its greatest unipotent image. Our main result uses the covering theorem of [8] to construct free weakly left ample monoids.

GRAPH EXPANSIONS OF RIGHT CANCELLATIVE MONOIDS

The relations ℛ* and on a monoid M are natural generalizations of Green’s relations ℛ and , which coincide with ℛ and if M is regular. A monoid M in which every ℛ*-class contains an idempotent is called left (right) abundant; if in addition the idempotents of M commute, that is, E(M) is a semilattice, then M is left (right) adequate. Regular monoids are obviously left (and right) abundant and inverse monoids are left (and right) adequate. Many of the well known results of regular and inverse semigroup theory have analogues for left abundant and left adequate monoids, or at least to special classes thereof. The aim of this paper is to develop a construction of left adequate monoids from the Cayley graph of a presentation of a right cancellative monoid, inspired by the construction of inverse monoids from group presentations, given by Margolis and Meakin in [10]. This technique yields in particular the free left ample (formerly left type A) monoid on a given set X.

PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT E -AMPLE SEMIGROUPS

We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation 7! C , where C is the identity map on the domain of . We investigate the construction of ‘actions’ from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schroder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

Fundamental Representations for Classes of Semigroups Containing a Band of Idempotents

The construction by Hall of a fundamental orthodox semigroup W B from a band B provides an important tool in the study of orthodox semigroups. Halls semigroup W B has the property that a semigroup is fundamental and orthodox with band of idempotents isomorphic to B if and only if it is embeddable as a full subsemigroup into W B . The aim of this article is to extend Halls approach to some classes of nonregular semigroups. From a band B, we construct a semigroup U B that plays the role of W B for a class of weakly B-abundant semigroups having a band of idempotents B. The semigroups we consider, in particular U B , must also satisfy a weak idempotent connected condition. We show that U B has subsemigroup V B where V B satisfies a stronger notion of idempotent connectedness, and is again the canonical semigroup of its kind. In turn, V B contains W B as its subsemigroup of regular elements. Thus we have the following inclusions as subsemigroups: either of which may be strict, even in the finite case. The existence of the semigroups U B and V B enable us to prove a structure theorem for classes of weakly B-abundant semigroups having band of idempotents B, and satisfying either of our idempotent connected conditions, as spined products of U B , or V B , with a weakly B/ 𝒟 -ample semigroup.

Enlargements, semiabundancy and unipotent monoids 1

The relation [Rtilde] on a monoid S provides a natural generalisation of Green’s relation R. If every [Rtilde]-class of S contains an idempotentS is left semiabundant; if [Rtilde] is a left congruence then S satisfies(CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy(CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples. This is the first of three related papers exploring the relationship between unipo-tent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with(CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence σ on a monoid S, we construct functors is a left adjoint of F σ. Here U is the category of unipotent monoids and F is a category of left semiabundant monoids with properties echoing those of F-inverse monoids.

RESTRICTION AND EHRESMANN SEMIGROUPS

Inverse semigroups form a variety of unary semigroups, that is, semi- groups equipped with an additional unary operation, in this case a 7! a 1 . The theory of inverse semigroups is perhaps the best developed within semigroup the- ory, and relies on two factors: an inverse semigroup S is regular, and has semilattice of idempotents. Three major approaches to the structure of inverse semigroups have emerged. Eectively, they each succeed in classifying inverse semigroups via groups (or groupoids) and semilattices (or partially ordered sets). These are (a) the Ehresmann-Schein-Nambooripad characterisation of inverse semigroups in terms of inductive groupoids, (b) Munns use of fundamental inverse semigroups and his construction of the semigroup TE from a semilattice E, and (c) McAlisters results showing on the one hand that every inverse semigroup has a proper (E-unitary) cover, and on the other, determining the structure of proper inverse semigroups in terms of groups, semilattices and partially ordered sets. The aim of this article is to explain how the above techniques, which were developed to study inverse semigroups, may be adapted for certain classes of bi-unary semi- groups. The classes we consider are those of restriction and Ehresmann semigroups. The common feature is that the semigroups in each class possess a semilattice of idempotents; however, there is no assumption of regularity.

Faithful functors from cancellative categories to cancellative monoids with an application to abundant semigroups

We prove that any small cancellative category admits a faithful functor to a cancellative monoid. We use our result to show that any primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup where M is a cancellative monoid and P is the identity matrix. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether a finite ample semigroup embeds as a full subsemigroup of an inverse semigroup.

RELATIVELY FREE ALGEBRAS WITH WEAK EXCHANGE PROPERTIES

We consider algebras for which the operation PC of pure closure of subsets satisfies the exchange property. Subsets that are independent with respect to PC are directly independent. We investigate algebras in which PC satisfies the exchange property and which are relatively free on a directly independent generating subset. Examples of such algebras include independence algebras and finitely generated free modules over principal ideal domains.