The translational hull of an adequate semigroup

1985 ◽  
Vol 32 (1) ◽  
pp. 79-86 ◽  
Author(s):  
John Fountain ◽  
Mark Lawson
Keyword(s):  
Adequate Semigroup

scholarly journals FREE ADEQUATE SEMIGROUPS

2011 ◽  
Vol 91 (3) ◽  
pp. 365-390 ◽  
Author(s):  
MARK KAMBITES
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Explicit Description ◽  
Finitely Generated ◽  
Free Objects ◽  
Ample Semigroup ◽  
Free Inverse Semigroup ◽  
Adequate Semigroup

AbstractWe give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial ‘folding’ operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are 𝒥-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.

scholarly journals Free right type A semigroups

1991 ◽  
Vol 33 (2) ◽  
pp. 135-148 ◽  
Author(s):  
John Fountain
Keyword(s):  
Binary Operation ◽  
Type A ◽  
Unary Operation ◽  
Adequate Semigroup ◽  
Green’S Relation

The relation ℒ* is defined on a semigroup S by the rule that a ℒ*b if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S) of idempotents is a subsemilattice of S. A right adequate semigroup is an E-semigroup in which every ℒ*-class contains an idempotent. It is easy to see that, in fact, each ℒ*-class of a right adequate semigroup contains a unique idempotent [8]. We denote the idempotent in the ℒ*-class of a by a*. Then we may regard a right adequate semigroup as an algebra with a binary operation of multiplication and a unary operation *. We will refer to such algebras as *-semigroups. In [10], it is observed that viewed in this way the class of right adequate semigroups is a quasi-variety.

scholarly journals Adequate Semigroups

1979 ◽  
Vol 22 (2) ◽  
pp. 113-125 ◽  
Author(s):  
John Fountain
Keyword(s):  
Identity Element ◽  
Adequate Semigroup ◽  
Green’S Relation ◽  
Left And Right ◽  
Special Classes

A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.

MATRIX REPRESENTATIONS OF AMPLE SEMIGROUPS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350108 ◽  
Author(s):  
XIAOJIANG GUO ◽  
K. P. SHUM
Keyword(s):  
Inverse Semigroups ◽  
Matrix Representations ◽  
Ample Semigroup ◽  
Adequate Semigroup ◽  
Brandt Semigroups

An adequate semigroup S is said to be ample if for any e2 = e, a ∈ S, ae = (ae)†a and ea = a(ea)*. It is well known that inverse semigroups are ample semigroups. The purpose of this paper is to study matrix representations of an ample semigroup. Some properties of ample semigroups are obtained. It is proved that any indecomposable good matrix representations of an ample semigroup can be constructed by using those of weak Brandt semigroups.

scholarly journals Proper left type-A monoids revisited

1993 ◽  
Vol 35 (3) ◽  
pp. 293-306 ◽  
Author(s):  
John Fountain ◽  
Gracinda M. S. Gomes
Keyword(s):  
Type A ◽  
Adequate Semigroup ◽  
Green’S Relation

The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.

scholarly journals Factorisable right adequate semigroups

1981 ◽  
Vol 24 (3) ◽  
pp. 171-178 ◽  
Author(s):  
Abdulsalam El-Qallali
Keyword(s):  
Type A ◽  
Adequate Semigroup ◽  
Green’S Relation

On a semigroup S the relation ℒ* is defined by the rule that (a, b) ∈ ℒ* if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S. It is well known that for a monoid S, every principal right ideal is projective if and only if each ℒ*-class of S contains an idempotent. Following (6) we say that a semigroup with or without an identity in which each ℒ*-class contains an idempotent and the idempotents commute is right adequate. A right adequate semigroup S in which eS ∩ aS = eaS for any e2 = e, a ∈ S is called right type A. This class of semigroups is studied in (5).

scholarly journals Locally adequate semigroup algebras

2016 ◽  
Vol 14 (1) ◽  
pp. 29-48 ◽  
Author(s):  
Yingdan Ji ◽  
Yanfeng Luo
Keyword(s):  
Direct Product ◽  
Simple Semigroup ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Special Cases ◽  
Direct Sum Decomposition ◽  
Adequate Semigroup

AbstractWe build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant $0{\rm{ - }}{\cal J}*$-simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the ${\cal R}*$-classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.

Semigroup algebras of finite ample semigroups

2012 ◽  
Vol 142 (2) ◽  
pp. 371-389 ◽  
Author(s):  
Xiaojiang Guo ◽  
Lin Chen
Keyword(s):  
Triangular Matrix ◽  
Inverse Semigroups ◽  
Semigroup Algebras ◽  
Matrix Representations ◽  
Ample Semigroup ◽  
Adequate Semigroup

An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.

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