Gracinda M. S. Gomes

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.

Nilpotents in finite symmetric inverse semigroups

In semigroup theory as in other algebraic theories a significant part of the total effort is appropriately applied to the study of certain standard examples occurring, as it were, “in nature”. The most obvious such semigroup is the full transformation semigroup ( X ) (see [ 3 ]) and about this semigroup a great deal is known in both the finite and infinite cases.

Proper left type- A monoids revisited

The relation ℛ* is defined on a semigroup S by the rule that ℛ* b if and only if the elements a, b of S are related by the Greens relation ℛ in some oversemigroup of S . A semigroup S is an E -semigroup if its set E(S) of idempotents is a subsemilattice of S . A left adequate semigroup is an E -semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a + .

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.

Almost Factorizable Weakly Ample Semigroups

An appropriate generalization of the notion of permissible sets of inverse semigroups is found within the class of weakly ample semigroups that allows us to introduce the notion of an almost left factorizable weakly ample semigroup in a way analogous to the inverse case. The class of almost left factorizable weakly ample semigroups is proved to coincide with the class of all (idempotent separating) (2, 1, 1)-homomorphic images of semigroups W(T, Y) where Y is a semilattice, T is a unipotent monoid acting on Y, and W(T, Y) is a well-defined subsemigroup in the respective semidirect product that appeared in the structure theory of left ample monoids more than ten years ago. Moreover, the semigroups W(T, Y) are characterized to be, up to isomorphism, just the proper and almost left factorizable weakly ample semigroups.

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT In this paper we calculate presentations for some natural monoids of transformations on a chain X n = {1 < 2 <⋅s < n}. First we consider 𝒪𝒟 n [𝒫𝒪𝒟 n ], the monoid of all full [partial] transformations on X n that preserve or reverse the order. Two other monoids of partial transformations on X n we look at are 𝒫𝒪𝒫 n and 𝒫𝒪ℛ n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.

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.

The Szendrei expansion of a semigroup

In the terminology of Birget and Rhodes [3], an expansion is a functor F from the category of semigroups into some special category of semigroups such that there is a natural transformation η from F to the identity functor for which η s is surjective for every semigroup S . The three expansions introduced in [3] have proved to be of particular interest when applied to groups. In fact, as shown in [4], Ĝ (2) are isomorphic for any group G , is an E -unitary inverse monoid and the kernel of the homomorphism η G is the minimum group congruence on . Furthermore, if G is the free group on A , then the “cut-down to generators” which is a subsemigroup of is the free inverse semigroup on A . Essentially the same result was given by Margolis and Pin [12].

Congruences on monoids of order-preserving or order-reversing transformations on a finite chain

This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain

A characterization of the group congruences on a semigroup.

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