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 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. 

SUBALGEBRAS OF THE SQUARES OF FINITE MINIMAL MAJORITY ALGEBRAS

2013 ◽  
Vol 23 (07) ◽  
pp. 1533-1549
Author(s):  
KALLE KAARLI
Keyword(s):  
Inverse Monoid ◽  
Abstract Characterization ◽  
The One ◽  
Majority Term ◽  
Matrix Construction

This paper provides an abstract characterization of the monoids that appear as monoids of subalgebras of the square of finite minimal algebras admitting a majority term. As a tool, a matrix construction similar to the one introduced in [A characterization of the inverse monoid of bi-congruences of certain algebras, Int. J. Algebra Comput.6 (2009) 791–808] is used.

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.

scholarly journals ON FREE SPECTRA OF A CLASS OF FINITE INVERSE MONOIDS

2012 ◽  
Vol 93 (3) ◽  
pp. 225-237
Author(s):  
IGOR DOLINKA
Keyword(s):  
Semidirect Product ◽  
Natural Order ◽  
Inverse Monoid ◽  
Left Action ◽  
Inverse Monoids ◽  
Group Of Units ◽  
Finite Monoids ◽  
Free Spectrum ◽  
Free Spectra

AbstractFor a finite Clifford inverse algebra $A$, with natural order meet-semilattice ${Y}_{A} $ and group of units ${G}_{A} $, we show that the inverse monoid obtained as the semidirect product ${ Y}_{A}^{1} {\mathop{\ast }\nolimits}_{\rho } {G}_{A} $ has a log-polynomial free spectrum whenever $\rho $ is a term-expressible left action of ${G}_{A} $ on ${Y}_{A} $ and all subgroups of $A$ are nilpotent. This yields a number of examples of finite inverse monoids satisfying the Seif conjecture on finite monoids whose free spectra are not doubly exponential.

GRAPH EXPANSIONS OF RIGHT CANCELLATIVE MONOIDS

1996 ◽  
Vol 06 (06) ◽  
pp. 713-733 ◽  
Author(s):  
VICTORIA GOULD
Keyword(s):  
Inverse Semigroup ◽  
Cayley Graph ◽  
Semigroup Theory ◽  
Type A ◽  
Inverse Monoids ◽  
Cancellative Monoid ◽  
Group Presentations ◽  
Free Left ◽  
Left And Right ◽  
Graph Expansions

The relations ℛ* and [Formula: see text] on a monoid M are natural generalizations of Green’s relations ℛ and [Formula: see text], which coincide with ℛ and [Formula: see text] if M is regular. A monoid M in which every ℛ*-class [Formula: see text] contains an idempotent is called left (right) abundant; if in addition the idempotents of M commute, that is, E(M) is a semilattice, then M is left (right) adequate. Regular monoids are obviously left (and right) abundant and inverse monoids are left (and right) adequate. Many of the well known results of regular and inverse semigroup theory have analogues for left abundant and left adequate monoids, or at least to special classes thereof. The aim of this paper is to develop a construction of left adequate monoids from the Cayley graph of a presentation of a right cancellative monoid, inspired by the construction of inverse monoids from group presentations, given by Margolis and Meakin in [10]. This technique yields in particular the free left ample (formerly left type A) monoid on a given set X.

scholarly journals Inverse semigroups determined by their partial automorphism monoids

2006 ◽  
Vol 81 (2) ◽  
pp. 185-198 ◽  
Author(s):  
Simon M. Goberstein
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Inverse Subsemigroups

AbstractThe partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.

scholarly journals Dual symmetric inverse monoids and representation theory

1998 ◽  
Vol 64 (3) ◽  
pp. 345-367 ◽  
Author(s):  
D. G. Fitzgerald ◽  
Jonathan Leech
Keyword(s):  
Representation Theory ◽  
Inverse Semigroup ◽  
The Other ◽  
Natural Order ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Dual Pairs ◽  
Representations Of Groups ◽  
Dual Symmetric Inverse Monoids ◽  
Symmetric Inverse Monoid

AbstractThere is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid Ix, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual Ix* to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in Ix and Ix*, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

Extensions of One Primitive Inverse Semigroup by Another

1972 ◽  
Vol 24 (2) ◽  
pp. 270-278
Author(s):  
Janet E. Ault
Keyword(s):  
Inverse Semigroup ◽  
Finite Number ◽  
Primitive Idempotent ◽  
Inverse Semigroups ◽  
Ideal Extension ◽  
Brandt Semigroup ◽  
Abstract Characterization

Every inverse semigroup containing a primitive idempotent is an ideal extension of a primitive inverse semigroup by another inverse semigroup. Consequently, in developing the theory of inverse semigroups, it is natural to study ideal extensions of primitive inverse semigroups (cf. [3; 7]). Since the structure of any primitive inverse semigroup is known, an obvious type of ideal extension to consider is that of one primitive inverse semigroup by another. In this paper, we will construct all such extensions and give an abstract characterization of the resulting semigroup.The problem of extending one primitive inverse semigroup by another can be essentially reduced to that of extending one Brandt semigroup by another Brandt semigroup. The latter problem has been solved by Lallement and Petrich in [3] in case the first Brandt semigroup has only a finite number of idempotents.

scholarly journals The Join of the Varieties of R-trivial and L-trivial Monoids via Combinatorics on Words

2012 ◽  
Vol Vol. 14 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Manfred Kufleitner ◽  
Alexander Lauser
Keyword(s):  
Finite Semigroup ◽  
Semigroup Theory ◽  
Combinatorics On Words ◽  
Finite Monoids ◽  
International Audience ◽  
Single Identity

Automata, Logic and Semantics International audience The join of two varieties is the smallest variety containing both. In finite semigroup theory, the varieties of R-trivial and L-trivial monoids are two of the most prominent classes of finite monoids. Their join is known to be decidable due to a result of Almeida and Azevedo. In this paper, we give a new proof for Almeida and Azevedo's effective characterization of the join of R-trivial and L-trivial monoids. This characterization is a single identity of omega-terms using three variables.

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