Minimum Group Congruence Research Articles

On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

In this paper we study the structure of the monoid Iℕn ∞ of cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞ is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U) by the symmetric group Sn.

The structure of certain F-inverse semigroups

An inverse semigroup is an F-inverse semigroup if each class modulo the minimum group congruence has a maximum element. This paper studies a class of F-inverse semi-groups in which all these maximum elements form a subgroup. After obtaining some characterizations of such semigroup, the structure theorem is obtained by the semidirect product of a semilattice and a group.

Minimum Group Congruences on Eventually Regular Semigroups

An eventually regular semigroup is a semigroup in which some power of any element is regular. The minimum group congruence on an eventually regular semigroup is investigated by means of weak inverse. Furthermore, some properties of the minimum group congruence on an eventually regular semigroup are characterized.

Zero-Semidistributive Inverse Semigroups

An inverse semigroup S is said to be 0-semidistributive if its lattice ℒF (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab) m = a n or (ab) m = b n , where σ is the minimum group congruence on S.

The Szendrei expansion of a semigroup

MathematikaVolume 37, Issue 2 p. 251-260 Research Article The Szendrei expansion of a semigroup John Fountain, John Fountain Department of Mathematics, University of York, Heslington, York, YO1 5DDSearch for more papers by this authorGracinda M. S. Gomes, Gracinda M. S. Gomes Departamento de Matemàtica, Faculdade de Ciências, Universidade de Lisboa, 1700 Lisboa, PortugalSearch for more papers by this author John Fountain, John Fountain Department of Mathematics, University of York, Heslington, York, YO1 5DDSearch for more papers by this authorGracinda M. S. Gomes, Gracinda M. S. Gomes Departamento de Matemàtica, Faculdade de Ciências, Universidade de Lisboa, 1700 Lisboa, PortugalSearch for more papers by this author First published: 01 December 1990 https://doi.org/10.1112/S0025579300012973Citations: 9AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume37, Issue2December 1990Pages 251-260 RelatedInformation

Properties of relatively free inverse semigroups

The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if S S is combinatorial (i.e., H \mathcal {H} is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, S S does not contain a copy of the bicyclic semigroup) or E E -unitary (i.e., E ( S ) E(S) is the kernel of the minimum group congruence) then the relatively free inverse semigroup F V X F{\mathcal {V}_X} on the set X X in the variety V \mathcal {V} generated by S S is also combinatorial, completely semisimple or E E -unitary, respectively. If S S is a fundamental (i.e., the only congruence contained in H \mathcal {H} is the identity congruence) and | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , then F V X F{\mathcal {V}_X} is also fundamental. F V X F{\mathcal {V}_X} may not be fundamental if | X | > ℵ 0 |X| > {\aleph _0} . It is also shown that for any variety of groups U \mathcal {U} and for | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , there exists a variety of inverse semigroups V \mathcal {V} which is minimal with respect to the properties (i) F V X F{\mathcal {V}_X} is fundamental and (ii) V ∩ G = U \mathcal {V} \cap \mathcal {G} = \mathcal {U} , where G \mathcal {G} is the variety of groups. In the main result of the paper it is shown that there exists a variety V \mathcal {V} for which F V X F{\mathcal {V}_X} is not completely semisimple, thereby refuting a long standing conjecture.

The minimum group congruence on an ℒunipotent semigroup

The minimum group congruence on an ℒunipotent semigroup

Proper Regular Semigroups

In a recent paper, D. B. McAlister gave several characterizations of proper inverse semigroups. In this paper, the concept of proper is extended to the class of regular semigroups. This is done by requiring that the set of idempotents of the semigroup coincides with the kernel of the minimum group congruence on the semigroup. A theorem is presented which contains several characterizations of proper regular semigroups, and the related result of McAlister then follows as a corollary. The paper concludes with some open questions and examples.

Proper regular semigroups

In a recent paper, D. B. McAlister gave several characterizations of proper inverse semigroups. In this paper, the concept of proper is extended to the class of regular semigroups. This is done by requiring that the set of idempotents of the semigroup coincides with the kernel of the minimum group congruence on the semigroup. A theorem is presented which contains several characterizations of proper regular semigroups, and the related result of McAlister then follows as a corollary. The paper concludes with some open questions and examples.

Strongly E-reflexive inverse semigroups

LetSbe an inverse semigroup with semilattice of idempotentsE. We denote by σ the minimum group congruence onS(6), and by τ the maximum idempotent-determined congruence onS(2). (Recall that the congruence η onSis calledidempotent-determinedif (e,x)∈ η ande∈Eimply thatx∈E.) In general τ ⊆ σ.

Embedding theorems for proper inverse semigroups

Let S be an inverse semigroup with semilattice of idempotents E, and let ϱ( S), or ϱ if there is no danger of ambiguity, be the minimum group congruence on S. Then S is said to be proper if Eϱ = E, or alternatively, if the equation ex = e for some e ϵ E and x ϵ S implies that x ϵ E. For example, free inverse semigroups and fundamental ω-inverse semigroups are proper. In a recent paper, McAlister has given a remarkable structure theorem for an arbitrary proper inverse semigroup, and using this theorem we show (Theorem 1.3) that any proper inverse semigroup P can be embedded in a semidirect product P ̄ of a semilattice and a group. Some consequences of this result are given; for example, if P is bisimple with identity then P ̄ is simple (see Theorem 1.6). Reilly has proved that an arbitrary inverse semigroup can be embedded in a bisimple inverse semigroup with identity. Given a proper inverse semigroup P it is shown that a proper inverse semigroup P′, arising from P by blowing up P g9 , can indeed be embedded in a bisimple proper inverse semigroup with identity (Theorem 2.4). We also investigate those homomorphic images of bisimple proper inverse semigroups which are themselves proper. In the final section, Rees quotient images of proper inverse semigroups are characterised (Theorem 3.4), and some other constructions for (simple) semidirect products of semilattices and groups are considered.

The lattice of inverse subsemigroups of a reduced inverse semigroup

An inverse semigroup R is said to be reduced (or proper) if ℛ∩σ= i (where σ is the minimum group congruence on R). McAlister has shown ([3], [4]) that every reduced inverse semigroup is isomorphic with a “P-semigroup” P(G, , ), for some semilattice , poset containing as an ideal, and group G acting on by order-automorphisms; (and, conversely, every P-semigroup is reduced). In [4], he also found the morphisms between P-semigroups, in terms of morphisms between the respective groups, and between the respective posets.

Inverse semigroups as extensions of semilattices

LetSbe an inverse semigroup with semilattice of idempotentsE, and letρbe a congruence onS. Thenρis said to beidempotent-determined[2], or I.D. for short, if (a, b) ∈ р anda∈Eimply thatb∈E. If, further,ρis a group congruence, then clearlyρis the minimum group congruence onS, and in this caseSis said to beproper[8]. LetT=S/ρ.

Proper Dubreil-Jacotin inverse semigroups

This paper is concerned mainly with the structure of inverse semigroups which have a partial ordering defined on them in addition to their natural partial ordering. However, we include some results on partially ordered semigroups which are of interest in themselves. Some recent information [1, 2, 6, 7,11] has been obtained about the algebraic structure of partially ordered semigroups, and we add here to the list by showing in Section 1 that every regular integrally closed semigroup is an inverse semigroup. In fact it is a proper inverse semigroup [10], that is, one in which the idempotents form a complete class modulo the minimum group congruence, and the structure of these semigroups is explicitly known [5].

Right group congruences on a semigroup

This paper develops necessary and sufficient conditions on an algebraic semigroup S in order that it will have nontrivial right group homomorphic images. A central notion used is that of how the normal subsemigroups associated with the group images of S relate to the right group images of S. The results presented thus extend those of R. R. Stoll. Where right group congruences exist, the structure of S is determined and the right group congruences are characterized in terms of group congruences and right zero congruences on S. Sufficient conditions are then found for the existence of a minimum right group congruence on S, and isomorphic right group congruences and the minimum right group congruence on S are described. Lastly, an application is made to regular semigroups whose idempotents form a rectangular band.

Certain fundamental congruences on a regular semigroup

In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by α β and η respectively.

The homology of principally directed ordered groupoids

We present some homological properties of a relation $\beta$ on ordered groupoids that generalises the minimum group congruence for inverse semigroups. When $\beta$ is a transitive relation on an ordered groupoid $G$, the quotient $G / \beta$ is again an ordered groupoid, and construct a pair of adjoint functors between the module categories of $G$ and of $G / \beta$. As a consequence, we show that the homology of $G$ is completely determined by that of $G / \beta$, generalising a result of Loganathan for inverse semigroups.