The minimum group congruence on an ℒunipotent semigroup

1979 ◽  
Vol 18 (1) ◽  
pp. 9-14 ◽  
Author(s):  
C. C. Edwards
Keyword(s):  
Group Congruence ◽  
Minimum Group

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

2019 ◽  
Vol 12 (3) ◽  
pp. 51-68
Author(s):  
Oleg Gutik ◽  
Anatolii Savchuk
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Natural Partial Order ◽  
Group Congruence ◽  
Partial Isometries ◽  
Group Of Units ◽  
Minimum Group ◽  
Positive Integers ◽  
Quotient Semigroup ◽  
Free Semilattice

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.

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

2019 ◽  
Vol 11 (2) ◽  
pp. 296-310
Author(s):  
O.V. Gutik ◽  
A.S. Savchuk
Keyword(s):  
Additive Group ◽  
Group Congruence ◽  
Topological Semigroups ◽  
Inverse Subsemigroup ◽  
Minimum Group ◽  
Identity Congruence ◽  
Positive Integers ◽  
Quotient Semigroup

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.

scholarly journals Proper Dubreil-Jacotin inverse semigroups

1975 ◽  
Vol 16 (1) ◽  
pp. 40-51 ◽  
Author(s):  
R. McFadden
Keyword(s):  
Inverse Semigroup ◽  
Algebraic Structure ◽  
Inverse Semigroups ◽  
Partial Ordering ◽  
Group Congruence ◽  
Complete Class ◽  
Ordered Semigroups ◽  
Partially Ordered ◽  
Minimum Group ◽  
Integrally Closed

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].

scholarly journals Minimum Group Congruences on Eventually Regular Semigroups

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-5
Author(s):  
Wang Yu ◽  
Yin ZhiXiang
Keyword(s):  
Regular Semigroup ◽  
Group Congruence ◽  
Regular Semigroups ◽  
Minimum Group ◽  
Eventually Regular Semigroup

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.

scholarly journals Certain congruences on a completely regular semigroup

1974 ◽  
Vol 15 (2) ◽  
pp. 109-120 ◽  
Author(s):  
Thomas L. Pirnot
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Congruence ◽  
Completely Regular Semigroup ◽  
Rees Matrix Semigroup ◽  
Semilattice Congruence ◽  
Completely Regular ◽  
Maximal Group ◽  
Minimum Group ◽  
Matrix Semigroup

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

scholarly journals Strongly E-reflexive inverse semigroups

1977 ◽  
Vol 20 (4) ◽  
pp. 339-354 ◽  
Author(s):  
L. O'Carroll
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Group Congruence ◽  
Minimum Group

Let S be an inverse semigroup with semilattice of idempotents E. We denote by σ the minimum group congruence on S (6), and by τ the maximum idempotent-determined congruence on S (2). (Recall that the congruence η on S is called idempotent-determined if (e, x)∈ η and e ∈ E imply that x ∈ E.) In general τ ⊆ σ.

scholarly journals Certain fundamental congruences on a regular semigroup

1966 ◽  
Vol 7 (3) ◽  
pp. 145-159 ◽  
Author(s):  
J. M. Howie ◽  
G. Lallement
Keyword(s):  
Inverse Semigroup ◽  
Algebraic Theory ◽  
Primary Concern ◽  
Group Congruence ◽  
Semilattice Congruence ◽  
Regular Semigroups ◽  
Recent Developments ◽  
Minimum Group ◽  
Band Congruence ◽  
Lattice Of Congruences

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.

scholarly journals Inverse semigroups as extensions of semilattices

1975 ◽  
Vol 16 (1) ◽  
pp. 12-21 ◽  
Author(s):  
Liam O'Carroll
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Group Congruence ◽  
Minimum Group

Let S be an inverse semigroup with semilattice of idempotents E, and let ρ be a congruence on S. Then ρ is said to be idempotent-determined [2], or I.D. for short, if (a, b) ∈ р and a∈E imply that b ∈ E. If, further, ρ is a group congruence, then clearly ρ is the minimum group congruence on S, and in this case S is said to be proper [8]. Let T = S/ρ.

scholarly journals The lattice of inverse subsemigroups of a reduced inverse semigroup

1976 ◽  
Vol 17 (2) ◽  
pp. 161-172 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Group Congruence ◽  
Minimum Group ◽  
Inverse Subsemigroups

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.

Group acceptance sampling plan for resubmitted lots based on life tests for odds exponential log logistic distribution

2017 ◽  
Vol 34 (8) ◽  
pp. 1343-1351 ◽  
Author(s):  
Rosaiah K. ◽  
Srinivasa Rao Gadde ◽  
Kalyani K. ◽  
Sivakumar D.C.U.
Keyword(s):  
Group Size ◽  
Sampling Plan ◽  
Logistic Distribution ◽  
Acceptance Sampling ◽  
Consumer’S Risk ◽  
Content Type ◽  
Acceptance Sampling Plan ◽  
Minimum Group ◽  
Termination Time ◽  
Group Acceptance

Purpose The purpose of this paper is to develop a group acceptance sampling plan (GASP) for a resubmitted lot when the lifetime of a product follows odds exponential log logistic distribution introduced by Rao and Rao (2014). The parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time. The authors compare the proposed plan with the ordinary GASP, and the results are illustrated with live data example. Design/methodology/approach The parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time. Findings The authors determined the group size and acceptance number. Research limitations/implications No specific limitations. Practical implications This methodology can be applicable in industry to study quality control. Social implications This methodology can be applicable in health study. Originality/value The parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time.

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