One-to-one partial right translations of a right cancellative semigroup

McAlister, D. B.

doi:10.1016/0021-8693(76)90158-7

Cited by 51 publications

(35 citation statements)

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“…In particular, if S is a regular semigroup, then SE = SB*. From (9) and (11) Proof. If (1) holds, then by definition each £C*-class and each 2ft*-class of S contains an idempotent.…”

Section: Preliminariesmentioning

confidence: 97%

Adequate Semigroups

Fountain

1979

Proceedings of the Edinburgh Mathematical Society

204

175

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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 X* which is defined as follows. On a semigroup S, (a,b)E.Z£* if and only if the elements a,b of S are related by Green's relation 2? in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each i£*-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 2?*-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 91* 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.Any inverse semigroup is adequate and in investigating adequate semigroups it is natural to look for analogues of results for inverse semigroups. In (6), Howie determines the greatest idempotent separating congruence n on an inverse semigroup S and shows that Slfi is isomorphic to the semilattice of idempotents of S if and only if the idempotents in S are central. In an adequate semigroup there need not be a greatest idempotent separating congruence. However, on an inverse semigroup, /x is the largest congruence contained in 7R,. Defining Stf* to be X* D91*, we determine, in Section 2, the largest congruence ix L contained in X* on a right adequate semigroup and hence find the largest congruence fi contained in "3t* on an adequate semigroup. We also find that the analogue of the second result of Howie mentioned above holds.It is well known (see (10) or (7, Chapter V)) that if S is an inverse semigroup with semilattice of idempotents E, then there is an idempotent separating homomorphism 0 from S into the Munn semigroup T E such that 0 °

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“…In particular, if S is a regular semigroup, then SE = SB*. From (9) and (11) Proof. If (1) holds, then by definition each £C*-class and each 2ft*-class of S contains an idempotent.…”

Section: Preliminariesmentioning

confidence: 97%

Adequate Semigroups

Fountain

1979

Proceedings of the Edinburgh Mathematical Society

204

175

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“…Alternative characterisations of the relation i? * are given by the following lemma from [14] and [17]. LEMMA …”

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confidence: 99%

Free right type A semigroups

Fountain

1991

Glasgow Math. J.

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Introduction. The relation i£* is defined on a semigroup 5 by the rule that a!£*b if and only if the elements a, b of S are related by Green's relation =2" in some oversemigroup of 5. A semigroup S is an E-semigroup if its set £(5) of idempotents is a subsemilattice of S. A right adequate semigroup is an £-semigroup in which every i?*-class contains an idempotent. It is easy to see that, in fact, each i?*-class of a right adequate semigroup contains a unique idempotent [8]. We denote the idempotent in the if-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.In this paper, which is the promised sequel to [10], we are concerned with right type A semigroups. These are semigroups which are right adequate and in which ea = a(ea)* for each a e S and e e E(S); they form a sub-quasi-variety of the quasi-variety of all right adequate semigroups. Thus, from general results in universal algebra, we know that free right type A semigroups exist. It is the purpose of this paper to give an explicit description of these free objects and to discuss some of their properties.Our approach is to make use of the construction of free right h-adequate semigroups in [10], a right adequate semigroup being right h-adequate when the mapping a a : E(Sy -> E(Sy defined by xa a = (xa)* is a homomorphism for each element a of 5. By a "-congruence on a right adequate semigroup 5, we mean a congruence on S regarded as a "-semigroup, that is, a semigroup congruence p on 5 which also satisfies apb implies a*pb*. A "-congruence p on a right adequate semigroup S is called a right type A congruence if Sip is a right type A semigroup, where the semigroup Sip is made into a "-semigroup by defining (ap)* to be a*p. On any right adequate semigroup, there is a minimum right type A congruence and if y is this congruence on P x , the free right h-adequate semigroup on X, then P x ly is the free right type A semigroup on X. For any non-empty set X, we construct a semigroup A x isomorphic to P x ly which is analogous to Scheiblich's construction [20] of the free inverse semigroup on X.This construction allows us to obtain several results which are analogues of theorems on inverse semigroups. For example, the fact that free inverse semigroups are E-unitary gives rise to one proof that every inverse semigroup has an £-unitary cover [18, Theorem VIII. 1.10]. The corresponding result for right type A semigroups is that every right type A semigroup has a proper cover [8]. On a right type A semigroup, the minimum left cancellative congruence is denoted by a and the semigroup is proper if a n S£* = t. It is easily seen that ^4^ is proper and we give a new proof of the covering result modelled on that in the inverse case.After recalling the basic properties of right type A semigroups in Section 1, we devote Section 2 to th...

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“…By a -rpp semigroup S we also mean that S is a semigroup which is -rpp for some ∈ LC(S). It is obvious that, if is the identical relation ε and R-equivalence on S, respectively, then L is exactly the relations L * and L * * on S stated in McAlister [19] and Tang [23]. In the sequel, we will identify the concepts of ε-rpp and R-rpp semigroups, respectively, with rpp and wrpp semigroups.…”

Section: Left C--rpp Semigroupsmentioning

confidence: 95%