scholarly journals Unit regular inverse monoids and Cliford monoids

2018 ◽  
Vol 7 (2.13) ◽  
pp. 306
Author(s):  
Sreeja V K
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Element ◽  
Inverse Monoid ◽  
Completely Regular Semigroup ◽  
Clifford Semigroup ◽  
Inverse Monoids ◽  
Completely Regular ◽  
Group Of Units

Let S be a unit regular semigroup with group of units G = G(S) and semilattice of idempotents E = E(S). Then for every there is a such that Then both xu and ux are idempotents and we can write or .Thus every element of a unit regular inverse monoid is a product of a group element and an idempotent. It is evident that every L-class and every R-class contains exactly one idempotent where L and R are two of Greens relations. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. A completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. A Clifford semigroup is a completely regular inverse semigroup. Characterization of unit regular inverse monoids in terms of the group of units and the semilattice of idempotents is a problem often attempted and in this direction we have studied the structure of unit regular inverse monoids and Clifford monoids. 

scholarly journals A characterization of Commutative Semigroups

2021 ◽  
Vol 12 (3) ◽  
pp. 5150-5155
Author(s):  
Dr. D. Mrudula Devi Et. al.
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Commutative Semigroup ◽  
Zero Semigroup ◽  
Completely Regular Semigroup ◽  
Semi Group ◽  
Commutative Semigroups ◽  
Completely Regular ◽  
Left Regular

This paper deals with some results on commutative semigroups. We consider (s,.) is externally commutative right zero semigroup is regular if it is intra regular and (s,.) is externally commutative semigroup then every inverse semigroup  is u – inverse semigroup. We will also prove that if (S,.) is a H -  semigroup then weakly cancellative laws hold in H - semigroup. In one case we will take (S,.) is commutative left regular semi group and we will prove that (S,.) is ∏ - inverse semigroup. We will also consider (S,.) is commutative weakly balanced semigroup  and then prove every left (right) regular semigroup is weakly separate, quasi separate and separate. Additionally, if (S,.) is completely regular semigroup we will prove that (S,.) is permutable and weakly separtive. One a conclusing note we will show and prove some theorems related to permutable semigroups and GC commutative Semigroups.

scholarly journals Generalized Neutrosophic Extended Triplet Group

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 327 ◽  
Author(s):  
Yingcang Ma ◽  
Xiaohong Zhang ◽  
Xiaofei Yang ◽  
Xin Zhou
Keyword(s):  
Regular Semigroup ◽  
Algebraic System ◽  
Classical Group ◽  
Algebra Structure ◽  
Completely Regular Semigroup ◽  
Single Factor ◽  
Clifford Semigroup ◽  
Completely Regular ◽  
Group 4

Neutrosophic extended triplet group is a new algebra structure and is different from the classical group. In this paper, the notion of generalized neutrosophic extended triplet group is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is a generalized neutrosophic extended triplet group if and only if it is a quasi-completely regular semigroup; (2) an algebraic system is a weak commutative generalized neutrosophic extended triplet group if and only if it is a quasi-Clifford semigroup; (3) for each n ∈ Z + , n ≥ 2 , ( Z n , ⊗ ) is a commutative generalized neutrosophic extended triplet group; (4) for each n ∈ Z + , n ≥ 2 , ( Z n , ⊗ ) is a commutative neutrosophic extended triplet group if and only if n = p 1 p 2 ⋯ p m , i.e., the factorization of n has only single factor.

scholarly journals Free completely regular semigroups

1984 ◽  
Vol 25 (2) ◽  
pp. 241-254 ◽  
Author(s):  
P. G. Trotter
Keyword(s):  
Regular Semigroup ◽  
Structure Theory ◽  
Completely Regular Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Additional Operation ◽  
Alternative Characterization ◽  
Completely Regular Semigroups

A completely regular semigroup is a semigroup that is a union of groups. The aim here is to provide an alternative characterization of the free completely regular semigroup Fcrx on a set X to that given by J. A. Gerhard in [3, 4].Although the structure theory for completely regular semigroups was initiated in 1941 [1] by A. H. Clifford it was not until 1968 that it was shown by D. B. McAlister [5] that Fcrx exists. More recently, in [7], M. Petrich demonstrated the existence of Fcrx by showing that completely regular semigroups form a variety of unary semigroups (that is, semigroups with the additional operation of inversion).

A CHARACTERIZATION OF THE INVERSE MONOID OF BI-CONGRUENCES OF CERTAIN ALGEBRAS

2009 ◽  
Vol 19 (06) ◽  
pp. 791-808 ◽  
Author(s):  
KALLE KAARLI ◽  
LÁSZLÓ MÁRKI
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Theory ◽  
Independent Interest ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Abstract Characterization ◽  
Finite Monoids ◽  
Ramsey’S Theorem ◽  
Matrix Construction

This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramsey's theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.

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 CENTRALISERS IN THE INFINITE SYMMETRIC INVERSE SEMIGROUP

2012 ◽  
Vol 87 (3) ◽  
pp. 462-479 ◽  
Author(s):  
JANUSZ KONIECZNY
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Green’S Relations ◽  
Completely Regular Semigroup ◽  
Completely Regular ◽  
Regular Elements ◽  
Green's Relations ◽  
Symmetric Inverse Semigroup ◽  
Alpha Beta

AbstractFor an arbitrary set $X$ (finite or infinite), denote by $\mathcal {I}(X)$ the symmetric inverse semigroup of partial injective transformations on $X$. For $ \alpha \in \mathcal {I}(X)$, let $C(\alpha )=\{ \beta \in \mathcal {I}(X): \alpha \beta = \beta \alpha \}$ be the centraliser of $ \alpha $ in $\mathcal {I}(X)$. For an arbitrary $ \alpha \in \mathcal {I}(X)$, we characterise the transformations $ \beta \in \mathcal {I}(X)$ that belong to $C( \alpha )$, describe the regular elements of $C(\alpha )$, and establish when $C( \alpha )$ is an inverse semigroup and when it is a completely regular semigroup. In the case where $\operatorname {dom}( \alpha )=X$, we determine the structure of $C(\alpha )$in terms of Green’s relations.

The Word Problem for Orthogroups

1981 ◽  
Vol 33 (4) ◽  
pp. 893-900 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Maximal Subgroup ◽  
Word Problem ◽  
Unary Operation ◽  
Group Inverse ◽  
Completely Regular Semigroup ◽  
Completely Regular ◽  
Image Position ◽  
Natural Way

A semigroup which is a union of groups is said to be completely regular. If in addition the idempotents form a subsemigroup, the semigroup is said to be orthodox and is called an orthogroup. A completely regular semigroup S is provided in a natural way with a unary operation of inverse by letting a-l for a ∈ S be the group inverse of a in the maximal subgroup of S to which a belongs. This unary operation satisfies the identities(1)(2)(3)In fact a completely regular semigroup can be defined as a unary semigroup (a semigroup with an added unary operation) satisfying these identities. An orthogroup can be characterized as a completely regular semigroup satisfying the additional identity(4)

Inverse semigroup homomorphisms via partial group actions

2001 ◽  
Vol 64 (1) ◽  
pp. 157-168 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Group Actions ◽  
Inverse Semigroups ◽  
Inverse Monoid ◽  
Partial Group ◽  
Inverse Monoids ◽  
Maximal Group ◽  
Special Cases ◽  
Quick Proof ◽  
Group Component

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

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