A note on the green's relation ℋ

1972 ◽  
Vol 23 (1-2) ◽  
pp. 105-108
Author(s):  
Ildikó Kiss
Keyword(s):  
Green’S Relation

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.

scholarly journals On residuated skew lattices

2019 ◽  
Vol 27 (1) ◽  
pp. 245-268
Author(s):  
Arsham Borumand Saeid ◽  
Roghayeh Koohnavard
Keyword(s):  
Congruence Relation ◽  
Residuated Lattice ◽  
Deductive System ◽  
Quotient Algebra ◽  
Skew Lattice ◽  
Skew Lattices ◽  
Green’S Relation

Abstract In this paper, we define residuated skew lattice as non-commutative generalization of residuated lattice and investigate its properties. We show that Green’s relation 𝔻 is a congruence relation on residuated skew lattice and its quotient algebra is a residuated lattice. Deductive system and skew deductive system in residuated skew lattices are defined and relationships between them are given and proved. We define branchwise residuated skew lattice and show that a conormal distributive residuated skew lattice is equivalent with a branchwise residuated skew lattice under a condition.

Green's Relation ℛ on the Monoid of Clone Endomorphisms

2005 ◽  
Vol 12 (03) ◽  
pp. 519-530 ◽  
Author(s):  
Thawhat Changphas ◽  
Klaus Denecke
Keyword(s):  
Green’S Relation ◽  
Natural Way

A hypersubstitution is a map which takes n-ary operation symbols to n-ary terms. Any such map can be uniquely extended to a map defined on the set Wτ(X) of all terms of type τ, and any two such extensions can be composed in a natural way. Thus, the set Hyp (τ) of all hypersubstitutions of type τ forms a monoid. In this paper, we characterize Green's relation ℛ on the monoid Hyp (τ) for the type τ=(n,n). In this case, the monoid of all hypersubstitutions is isomorphic with the monoid of all clone endomorphisms. The results can be applied to mutually derived varieties.

Weak inverses modulo Green’s relation $$\mathcal{{H}}$$ H on E-inversive and group-closed semigroups

2015 ◽  
Vol 92 (3) ◽  
pp. 691-711 ◽  
Author(s):  
Xingkui Fan ◽  
Qianhua Chen ◽  
Xiangjun Kong
Keyword(s):  
Green’S Relation

THE ORBIT STRUCTURE OF FINITE MONOIDS

1993 ◽  
Vol 03 (04) ◽  
pp. 557-573 ◽  
Author(s):  
ROB CARSCADDEN
Keyword(s):  
Transformation Semigroup ◽  
Unit Group ◽  
Full Transformation ◽  
Finite Monoid ◽  
Orbit Structure ◽  
Finite Monoids ◽  
Full Transformation Semigroup ◽  
Green’S Relation

Let M be a finite monoid with unit group G. We consider a refinement, [Formula: see text] of the Green’s relation [Formula: see text]. The [Formula: see text]-classes, denoted [Formula: see text] are the G×G orbits, GHG, of the ℋ-classes, H, of M. With an orbit [Formula: see text] we associate a local monoid [Formula: see text] and determine the structure of these local monoids. The theory is applied to the full transformation semigroup [Formula: see text] and we see that the number of orbits [Formula: see text] in [Formula: see text] is equal to the number of partitions of n.

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+.

Sign in / Sign up

Export Citation Format

Share Document