Some covering theorems for locally inverse semigroups

McAlister, D. B.

doi:10.1017/s1446788700022175

Cited by 6 publications

(9 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…11.1; [32], section 8.1) and latterly the re ‡ection monoids of Everitt and Fountain [8]. Appropriate generalisations of the concept were developed: signi…cantly, Lawson [17] identi…ed the appropriate generalisation from monoids to semigroups as almost factorizable semigroups, which had been used in McAlister [25]; for an account, see section 7.1 of [18]. Tirasupa [35] examined the Cli¤ord by semilattice case and Mills [29] the group by aperiodic case.…”

Section: Some Historymentioning

confidence: 99%

Factorizable inverse monoids

FitzGerald

2009

Semigroup Forum

View full text Add to dashboard Cite

Factorizable inverse monoids constitute the algebraic theory of those partial symmetries which are restrictions of automorphisms; the formal de…nition is that each element is the product of an idempotent and an invertible. This class of monoids has theoretical signi…cance, and includes concrete instances which are important in various contexts. This survey is organised around the idea of group acts on semilattices and contains a large range of examples. Topics also include methods for construction of factorizable inverse monoids, and aspects of their inner structure, morphisms, and presentations.

show abstract

Section: Some Historymentioning

confidence: 99%

Factorizable inverse monoids

FitzGerald

2009

Semigroup Forum

View full text Add to dashboard Cite

show abstract

“…In this section, we investigate the relationship between arbitrary inverse semigroups and almost factorisable semigroups. In particular, we shall make precise an embedding first obtained by McAlister [3], [5]. DEFINITION.…”

Section: / > Is Injective For Suppose That Y(p)=ip(q)mentioning

confidence: 99%

“…We were led to formulate them by considering the semigroup analogue of Ehresmann's Maximum Enlargement Theorem which is discussed in [1]. Conditions (El) and (E2) were mentioned by McAlister who calls subsemigroups satisfying these conditions alone heavy [3], [5]. The final condition, (E3), is mentioned in a remark in [3, Section 6.1.2], but is not used in the paper as a whole.…”

Section: I(s)) Then X E I{s) (E3) For Each E E E(t) There Exists / Ementioning

confidence: 99%

“…Theorem 3.6 of [3] is a special case. Then with componentwise multiplication P is an E-unitary semigroup over a semilattice, n t :P-*S is a surjective, idempotent separating homomorphism, and K 2 :P-*G is a surjective, iE-injecdve homomorphism.…”

Section: Let T Be An Enlargement Of S and Let 6 Be A Surjective Idempmentioning

confidence: 99%

“…Introduction. In [3], McAlister introduced a class of semigroups, called covering semigroups, which were shown to play an important role in the theory of E-unitary covers of semigroups. Strangely, this class of semigroups appears to have received little attention subsequently.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Almost factorisable inverse semigroups

Lawson

1994

Glasgow Math. J.

View full text Add to dashboard Cite

0. Introduction. In [3], McAlister introduced a class of semigroups, called covering semigroups, which were shown to play an important role in the theory of E-unitary covers of semigroups. Strangely, this class of semigroups appears to have received little attention subsequently. It is the aim of this paper to rehabilitate them and to study their properties in more detail. As a first step, we have chosen to rename them almost factorisable semigroups, since they can be regarded as the semigroup analogues of factorisable inverse monoids. Before discussing the contents of this paper in more detail we recall some standard terminology.We assume the reader is familiar with basic inverse semigroup theory. For convenience, however, we describe here the notation and terminology used which is not explicitly defined in the text. All semigroups will be inverse. The (generalised) inverse of an element x will be denoted by x~\ The set of idempotents of a semigroup is written £(S); if / I c 5 then E(A) = A (1 E(S). The natural partial order is denoted by « . A semigroup 5 is E-unitary if e ^x and e e E(S) implies x e £(5). An E-unitary cover of an inverse semigroup S is an E-unitary semigroup P and a surjective, idempotent separating homomorphism d:P-*S. It is well-known that every inverse semigroup has an E-unitary cover [7]. If 5 is a monoid then the group of units of 5 will be denoted by U(S). An inverse monoid S is said to be factorisable if for each s eS there exists g e U(S) such that s ^g. The minimum group congruence o on a semigroup S is denned by xoy if, and only if, there exists z e S such that z < x and z^y. If (Q, £ ) is a poset then A c Q is said to be an order ideal if b < a eA implies that b eA. The set [a] = {b e Q:b <«} is called a principal order ideal. The trace product of x and y in a semigroup 5 is defined to be xy if x~lx =yy~l, and undefined otherwise. If A and B are sets then the projections from A x B to A and B respectively are denoted by n x and jt 2 . If 6:A^B is a function then ker 6 denotes the equivalence relation induced on A. Finally, if p is a congruence then p* denotes the corresponding natural map.The paper is divided into three sections. In Section 1, we discuss the basic properties of almost factorisable inverse semigroups and show that they are very closely related to factorisable inverse monoids. Specifically, we prove in Proposition 1.7 that an inverse monoid is almost factorisable if, and only if, it is factorisable. In Theorem 1.10, we show that for every almost factorisable inverse semigroup S there is a factorisable inverse monoid F such that S is isomorphic to F\U(F). In Section 2, we describe all strongly E-unitary covers of almost factorisable inverse semigroups. This is achieved in Proposition 2.3 and Theorem 2.5. In Theorem 2.4, we show that the class of almost factorisable semigroups is the closure under homomorphisms of the class of strongly E-unitary semigroups. In the final section, Section 3, we relate arbitrary inverse semigroups to almost factorisable inverse semigroups...

show abstract

On Locally Inverse Semigroups Having Weakly E-unitary Covers

Szendrei

2004

Semigroup Forum

View full text Add to dashboard Cite

Some covering theorems for locally inverse semigroups

Cited by 6 publications

References 15 publications

Factorizable inverse monoids

Factorizable inverse monoids

Almost factorisable inverse semigroups

On Locally Inverse Semigroups Having Weakly E-unitary Covers

Contact Info

Product

Resources

About