Amenability of semigroups and their algebras modulo a group congruence

We investigate the amenability of the semigroup algebras \({\ell^1(S/\rho)}\) , where \({\rho}\) is a group congruence (not necessarily minimal) on a semigroup S. We relate this to a new notion of amenability of Banach algebras modulo an ideal, to prove a version of Johnson’s theorem for a large class of semigroups, including inverse semigroups, E-inversive semigroup and E-inversive E-semigroups.     

On Classes of E-Inversive Semigroups and Semigroups Whose Idempotents Form a Subsemigroup

Abstract

A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.     

E-inversive semigroups with a completely simple kernel

We show that an E-inversive semigroup S has a completely simple kernel K_S if and only if it contains a primitive idempotent (in that case, K_S is the set-theoretic union of the groups eSe, where e is a primitive idempotent of S). Along the way, some equivalent conditions for a semigroup to be E-inversive are given. Moreover, some applications of the above theorem will be pointed out.     

<Emphasis Type="Italic">E</Emphasis>-inversive Dubreil-Jacotin semigroups

Using group congruences, we obtain necessary and sufficient conditions for an ordered E-inversive semigroup to be a Dubreil-Jacotin semigroup. We also determine when such a semigroup is naturally ordered. In particular, when the subset of regular elements is a subsemigroup it contains a multiplicative inverse transversal.     

Ω-compact semigroups having congruence extension property

The congruence extension property (CEP) of semigroups has been extensively studied by a number of authors. We call a compact semigroup S an Ω-compact semigroup if the set of all regular elements of S forms an ideal of S. In this note, we characterize the Ω-compact semigroup having (CEP). Our result extends a recent result obtained by X.J. Guo on the congruence extension property of strong Ω-compact semigroups which is a semigroup containing precisely one regular D-class.     

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ? a congruence on S. Consider the sequence ? ⁽⁰⁾?? ⁽¹⁾???? ⁽ⁿ⁾?? of congruences on S, where ? ⁽⁰⁾=? and, for an arbitrary non-negative integer n, ? ⁽ⁿ⁺¹⁾ is defined by (a,b)∈? ⁽ⁿ⁺¹⁾ if and only if (xa,xb)∈? ⁽ⁿ⁾ for all x∈S. We show that $\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )$ for an arbitrary congruence ? on a semigroup S, where lrc(?) denotes the least left reductive congruence on S containing ?. We focuse our attention on congruences ? on semigroups S for which the congruence $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is left reductive. We prove that, for a congruence ? on a semigroup S, $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is a left reductive congruence of S if and only if $\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}$ is a left reductive congruence on the factor semigroup S/? (here ι _(S/?) denotes the identity relation on S/?). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S ⁿ =S ⁿ⁺¹ is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.     

N(2,2,0)代数的E-反演半群

本文研究了N(2,2,0)代数(S,*,△,0)的E-反演半群.利用N(2,2,0)代数的幂等元,弱逆元,中间单位元的性质和同宇关系,得到了N(2,2,0)代数的半群(S,*)构成E-反演半群的条件及元素α的右伴随非零零因子唯一,且为α的弱逆元等结论,这些结果进一步刻画了N(2,2,0)代数的结构.     

The VT-operator Semigroup for Two Kinds of Regular Semigroups

Let S be a regular semigroup and Con S the congruence lattice of S. For every rho element of Con S there exists a greatest congruence rhoV [smallest congruence rhov] on S such that the idempotent (rhoV/rho)-classes [(rho/rhov)-classes] are rectangular bands, and a greatest congruence rhoT [smallest congruence rhot] on S such that the idempotent (rhoT/rho-classes [(rho/rhot-classes] are groups. The subsemigroup of the transformation semigroup on Con S generated by the transformation rho → rhoV, rho → rhov, rho → rhoT, and rho → rhot, rho element of Con S, is investigated for orthodox semigroups and cryptogroups. It is shown that in this case this so-called Vt-operator semigroup Omega(S) contains 17 elements at most. A 17-element Vt-operator semigroup Omega(F) is realized for some regular orthogroup F.     

The numerical duplication of a numerical semigroup

In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup S and a semigroup ideal E?S, produces a new numerical semigroup, denoted by S? ^b E (where b is any odd integer belonging to S), such that S=(S? ^b E)/2. In particular, we characterize the ideals E such that S? ^b E is almost symmetric and we determine its type.     

A Representation for $F$-Regular Semigroups

A regular (inverse) semigroup S is called F-regular (F-inverse), if each class of the least group congruence _S contains a greatest element with respect to the natural partial order on S. Such a semigroup is necessarily an E-unitary regular (hence orthodox) monoid. We show that each F-regular semigroup S is isomorphic to a well determined subsemigroup of a semidirect product of a band X by S/_S, where X belongs to the band variety, generated by the band of idempotents E_S of S. Our main result, Theorem 4, is the regular version of the corresponding fact for inverse semigroups, and might be useful to generalize further features of the theory of F-inverse semigroups to the F-regular case.     

Ordered groupoid quotients and congruences on inverse semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s \({\mathcal {J}}\)–relation. The corresponding equivalence relation \(\simeq _N\) is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.     

Biordered sets and fundamental semigroups

Given any biordered set E, a natural construction yields a semigroup T _E that is always fundamental, in the sense that T _E possesses no nontrivial idempotent-separating congruence. In the case that E=E(S) is the biordered set of idempotents of a semigroup S generated by regular elements, there is a natural representation of S by T _E , such that S becomes a biorder-preserving coextension of a fundamental and symmetric subsemigroup of T _E . If further S is regular then this yields the fundamental constructions of Nambooripad, Grillet and Hall, which in turn generalise the construction of Munn of a maximum fundamental inverse semigroup from its semilattice of idempotents.     

A Characterization of Semirings Which Are Subdirect Products of a Distributive Lattice and a Ring

E -inversive semiring and a Clifford semiring and show that a semiring S is a subdirect product of a distributive lattice and a ring if and only if S is an E-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, as a consequence of these results we extend a result of Galbiati and Veronesi [2] in the case of Boolean semirings.     

On right congruences of semigroups having no proper essential right congruences

A right congruence ?? in a semigroup S is essential if for any right congruence ?? we have ??????=?? (the identity relation) implies ??=??. Clearly, the universal relation, ??, is an essential right congruence. We say ?? is proper if ??????. In this paper we get a necessary and sufficient condition for a semigroup with an identity element?1 and having no proper essential right congruences to have a distributive lattice of right congruences.     

On algebraic semigroups and monoids,II

Consider an algebraic semigroup S and its closed subscheme of idempotents, E(S). When S is commutative, we show that E(S) is finite and reduced; if in addition S is irreducible, then E(S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E(S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E(S) is smooth, and its connected components are conjugacy classes of the unit group.