On semigroups admitting ring structure II

We shall consider semigroups with O, which contain a unique maximal right ideal generated by a finite number of independent generators and in which every proper right ideal is contained in the unique maximal right ideal and investigate when these semigroups are multiplicative semigroups of a ring. We prove in particular that the necessary condition for this class of semigroups S to admit ring structure is S=S² if |S|>2. Furthermore the admissible ring structure of S is determined when the product of every two generators of the maximal right ideal M is O and when S satisfies one of the two conditions, namely S is commutative without idempotents except O and 1 or every generator of M is nilpotent.     

On Partially Ordered Semigroups and an Abstract Set-Difference

In this paper we prove the independence of a system of five axioms (S1)–(S5), which was proposed in the book of Pallaschke and Urbański (Pairs of Compact Convex Sets, vol. 548, Kluwer Academic Publishers, Dordrecht, 2002) for partially ordered commutative semigroups. These five axioms (S1)–(S5) are stated in the introduction below. A partially ordered commutative semigroup satisfying these axioms is called a F-semigroup. By the use of a further axiom (S6) we define an abstract difference for the elements of a F-semigroup and prove some basic properties. The most interesting example of a F-semigroup are the nonempty compact convex sets of a topological vector space endowed with the Minkowski sum as operation and the inclusion as partial order. In Section 4 we apply the abstract difference to the problem of minimality of convex fractions. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Semigroups with the Congruence Extension Property

ρT on a semigroup of T of S extends to the semigroup S, if there exists a congruence _ρ on s such that _{ρ|T= ρT}. A semigroup S has the congruence extension property, CEP, if each congruence on each semigroup extends to S. In this paper we characterize the semigroups with CEP by a set of conditions on their structure (by this we answer a problem put forward in [1]). In particular, every such semigroup is a semilattice of nil extensions of rectangular groups.     

Finite union of commutative power joined semigroups

A commutative semigroup is called power joined if for every element a, b there are positive integers m, n such that a^m=bⁿ. A commutative power joined semigroup is archimedean (p. 131, [3]) and cannot be decomposed into the disjoint union of more than one subsemigroup. Every commutative semigroup is uniquely decomposed into the disjoint union of power joined subsemigroups which are called the power joined components. This paper determines the structure of commutative archimedean semigroups which have a finite number of power joined components. The number of power joined components of commutative archimedean semigroups is one or three or infinity. The research for this paper was supported in part by NSF Grant GP-11964.     

Commutative semigroups whose homomorphic images in groups are groups

In this paper we study commutative semigroups whose every homomorphic image in a group is a group. We find that for a commutative semigroup S, this property is equivalent to S being a union of subsemigroups each of which either has a kernel or else is isomorphic to one of a sequence T₀, T₁, T₂, ... of explicitly given, countably infinite semigroups without idempotents. Moreover, if S is also finitely generated then this property is equivalent to S having a kernel.     

On the categoricity of semigroup-theoretical properties

By asemigroup-theoretical property we mean a property of semigroups which is preserved by isomorphism. Such a property iscategorical if it can be expressed in the language of categories: roughly, without using elements. We show that this is always possible with the proviso that in the case of one-sided properties we cannot refer in categorical terms to a specific side. For example, the property of having aleft identity cannot be described categorically in the category of semigroups, since the functor ()^op which takes a semigroup into its “opposite” semigroup is a category automorphism. We show that ()^op is the only non-trivial automorphism of the category of semigroups (up to natural equivalence of functors). In other words, the “automorphism group” of the category of semigroups has order two.     

Bibliographical Items

The H×H-Theorem. If S is a compact connected monoid with group of units H and with E(S) = {0,1}, and if S/(H×H) (the space of orbits HsH) has a total order defning the quotient topology, then there is a one parameter semigroups I with E(I)=E(S) which commutes elementwise with H. (In particular the function (h, i)→hi∶H×I→HI=S is a surmorphism, and S is cylindrical.) This is Theorem VI in Elements of Compact Semigroups, by Hofmann and Mostert (p. 177). H. Carruth discovered a gap in the proof of this theorem in 1971. The methods of proof presented here differ from theose originally suggested and do not use peripherality. byt do use transformation group theory, and the authors' earlier results (Semigroup Forum 3 (1972), 31–42). The H×H-theorem is generalized to yield a theorem which belongs to the context of Theorem VIII in the Elements (p. 204):Theorem: Let S be a compact monoid such that the orbit space S/(H×H) is a totally ordered connected space with M(S) as its minimal point. If all regular D-classes are subsemigroups, then there is an I-semigroup with E(I)=E(S) which commutes elementwise with H. (In particular S=HI as in the H×H theorem). The sufficient condition about the regular D-classes is clearly necessary.). Further sample result:Theorem. IfH is a congruence in a compact connected monoid, with zero, then the centralizer of the group of units is connected.     

Embedding topological semigroups in topological groups

In [6] Rothman investigated the problem of embedding a topological semigroup in a topological group. He defined a concept calledProperty F and showed that Property F is a necessary and sufficient condition for embedding a commutative, cancellative topological semigroup in its group of quotients as an open subset. This paper announces a generalization of Rothman’s result by definingProperty E and stating that a completely regular topological semigroup S can be embedded in a topological group by a topological isomorphism if and only if S can be embedded (algebraically) in a group and S has Property E. Property E is defined by first constructing a free topological semigroup (Theorem 1.1). This construction resembles the one in [4] for a free topological group. Full details, examples, and other embedding results will appear elsewhere. Some of the results in this paper were contained in the author’s doctoral dissertation written at Rutgers University under Professor Louis F. McAuley.     

Implicative commutative semigroups are equivalent to a class of BCK algebras

Chan and Shum [2] introduced the notion of implicative semigroups and obtained some of its important properties. BCK algebras with condition (S) were introduced by Iséki [4] and extensively investigated by several authors. In this note, we prove that implicative commutative semigroups are equivalent to BCK algebras with condition (S), that is, given an algebra <S;≤,·,*,1> of type (2,2,0), define ⊗ by stipulatingx⊗y=y*x and ≺ by puttingx≺y if and only ify≤x, then <S≤,·,*,1> is an implicative commutative semigroup if and only if <S;≺,·,⊗, 1> is a BCK algebra with condition (S); a nonempty subsetF ofS is an ordered filter of <S;≤,·,*, 1> if and only ifF is an ideal of <S;≺,·, ⊗, 1>. The author would like to thank the referee for his valuable comments which helped in the modification of this paper.     

The structure of commutative semigroups with the ideal extension property

In this paper, we will characterize commutative semigroups which have the ideal extension property (IEP). This characterization describes the multiplicative structure of commutative semigroups with IEP. Establishing this characterization was motivated not only by an interest in IEP itself, but also by the fact that in the category of commutative semigroups, the congruence extension property (CEP) implies IEP. A few preliminary results which hold in the general (non-commutative) case are discussed below. Following these initial observations, all semigroups considered are commutative.     

Finite semigroups with slowly growing pn-sequences

The p _n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p _n (S) ≤cn ^r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p _n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p _n -sequence and give some examples. Received March 5, 1999; accepted in final form November 1, 1999.     

A New Graph Structure of Commutative Semigroups

In this paper, a new zero-divisor graph $\overline{\G}(S)$ is defined and studied for a commutative semigroup $S$ with zero element. The properties and the structure of the graph are studied; for any complete graph and complete bipartite graph $G$, commutative semigroups $S$ are constructed such that the graph $G$ is isomorphic to $\overline{\G}(S)$.     

On pseudo-amenability of commutative semigroup algebras and their second duals

In this paper, we first characterize pseudo-amenability of semigroup algebras \(\ell ^1(S),\) for a certain class of commutative semigroups S,  the so-called archimedean semigroups. We show that for an archimedean semigroup S,  pseudo-amenability, amenability and approximate amenability of \(\ell ^1(S)\) are equivalent. Then for a commutative semigroup S,  we show that pseudo-amenability of \(\ell ^{1}(S)\) implies that S is a Clifford semigroup. Finally, we give some results on pseudo-amenability and approximate amenability of the second dual of a certain class of commutative semigroup algebras \(\ell ^1(S)\).     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

On the congruence extension property for compact semigroups

A topological semigroupS is said to have thecongruence extension property (CEP) provided that for each closed subsemigroupT ofS and each closed congruence σ onT, σ can be extended to a closed congruence onS. (That is, ∩(T xT=σ). The main result of this paper gives a characteriation of Γ-compact commutative archimedean semigroups with the congruence extension property (CEP). Consideration of this result was motivated by the problem of characterizing compact commutative semigroups with CEP as follows. It is well known that every commutative semigroup can be expressed as a semilattice of archimedean components each of which contains at most one idempotemt. The components of a compact commutative semigroup need not be compact (nor Γ-compact) as the congruence providing the decomposition is not necessarily closed. However, any component with CEP which is Γ-compact is characterized by the afore-mentioned result. Characterization of components of a compact commutative semigroup having CEP is a natural step towar characterization of the entire semigroup since CEP is a hereditary property. Other results prevented in this paper give a characterization of compact monothetic semigroups with CEP and show that Rees quotients of compact semigroups with CEP retain CEP.     

Definability in the lattice of equational theories of commutative semigroups

In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Jezek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular, that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and that such lattices have no automorphisms besides the obvious syntactically defined ones (with exceptions for special unary types). He has proved also that the most important classes of theories of a given type are so definable. In a later paper, Jezek and McKenzie have ``almost proved" the same facts for the lattice of equational theories of semigroups. There were good reasons to believe that the same can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, we show that the case of commutative semigroups is different.

     

关于弱交换PO—半群中的一个问题

本文引入弱交换ｐｏ－半群的概论２，研究这类半群到Ａｒｃｈｉｍｅｄｅａｎ子半群的半格分解，得到了这半群类似于具平凡序的弱交换半群的一个特征，由此在更一般的情形下回答了Ｋｅｈａｙｏｐｕｌｕ在「１」中提出的一个问题，并作为推论得到弱交换ｐｏｅ－半群和具平凡序的弱交换半群的已知结果。     

On semigroups in which every Rees one-sided congruence is a congruence

The purpose of this paper is to examine the structure of those semigroups which satisfy one or both of the following conditions: A_r(A_ℓ): The Rees right (left) congruence associated with any right (left) ideal is a congruence. The conditions A_r and A_ℓ are generalizations of commutativity for semigroups. This paper is a continuation of the work of Oehmke [5] and Jordan [4] on H-semigroups (H for hamiltonian, a semigroup is called an H-semigroup if every one-sided congruence is a two-sided congruence). In fact the results of section 2 of Oehmke [5] are proved here under the condition A_r and/or A_ℓ and not the stronger hamiltonian condition. Section 1 of this paper is essentially a summary of the known results of Oehmke. In section 2 we examine the structure of irreducible semigroups satisfying the condition A_r and/or A_ℓ. In particular we determine all regular (torsion) irreducible semigroups satisfying both the conditions A_r and A_ℓ. This research has been supported by Grant A7877 of the National Research Council of Canada.