On a Class of Subdirect Products of Left and Right Clifford Semigroups

Abstract

Regular semigroups S with the property eS ? Se or Se ? eS for all idempotents e ∈ S include all left and right Clifford semigroups. Characterizations of such semigroups are given and their structure investigated, in particular in terms of spined products of left and right Clifford semigroups with respect to Clifford semigroups.     

Principal Weak Flatness and Regularity of Diagonal Acts

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ? left P(P) ? weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 _X , D(S) is regular.     

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.     

On Some Monogenic Semigroups with an Associate Subgroup

Let S be any semigroup and a, s ∈ S. If a = asa, then s is an associate of a. A subgroup G of S is an associate subgroup of S if every a ∈ S has a unique associate a* in G. It turns out that G = H _z for some idempotent z, the zenith of S. The mapping a → a* is a unary operation on S. We say that S is monogenic if S is generated, as a unary semigroup, by a single element.

We embark upon the problem of the structure of monogenic semigroups in this sense by characterizing monogenic ones belonging to completely simple semigroups, normal cryptogroups, orthogroups, combinatorial semigroups, cryptic medial semigroups, cryptic orthodox semigroups, and orthodox monoids. In each of these cases, except one, we construct a free object. The general problem remains open.     

Flatness properties of diagonal acts over monoids

If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S ^I (for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness. Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats, etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems. S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494. Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007).     

Absolute flatness of regular semigroups with a finite height function

In this paper we give a sufficient condition for regular semigroups with a finite height function to be left absolutely flat. As a consequence, we can show that the semigroup Λ(S) of all right translations of a primitive regular semigroupS with only finitely manyR-classes, with composition being from left to right, is absolutely flat and give a generalization of a Bulman-Fleming and McDowell result concerning absolutely flat semigroups from primitive regular semigroups to regular semigroups with a finite height function. These results give examples of semigroups which are amalgamation bases in the class of semigroups. The author thanks the referee for finding errors in the original version of this paper.     

Intuitionistic fuzzy semiprime ideals in ordered semigroups

We give characterizations of different classes of ordered semigroups by using intuitionistic fuzzy ideals. We prove that an ordered semigroup is regular if and only if every intuitionistic fuzzy left (respectively, right) ideal of S is idempotent. We also prove that an ordered semigroup S is intraregular if and only if every intuitionistic fuzzy two-sided ideal of S is idempotent. We give further characterizations of regular and intra-regular ordered semigroups in terms of intuitionistic fuzzy left (respectively, right) ideals. In conclusion of this paper we prove that an ordered semigroup S is left weakly regular if and only if every intuitionistic fuzzy left ideal of S is idempotent.     

Right P-Comparable Semigroups

In this article we utilise the notion of right waist and right comparizer to study the ideal theory of semigroups. We also consider which of the properties of right cones can be carried over to right P-comparable semigroups. We give sufficient and necessary conditions on the set of nilpotent elements of a semigroup to be an ideal, and we provide several equivalent characterizations for a right ideal being a right waist. In one of our main results we show that in a right P ₁-comparable semigroup with left cancellation law, a prime segment P ₂ ? P ₁ is Archimedean, simple or exceptional. This extends a similar result pertaining to right cones.     

Unary Semigroups with an Associate Subgroup

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S.     

ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J ^*-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H ^* is a congruence on S such that S/H ^* is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J ^*-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.     

E-minimal semigroups

TheE-minimal semigroups are completely characterized. It turns out that there are four classes ofE-minimal semigroups:p-groups, left (right) zero semigroups, nilpotent semigroups, and the two element semilattice.     

Largest subsemigroups of the full transformation monoid

In this paper we are concerned with the following question: for a semigroup S, what is the largest size of a subsemigroup T?S where T has a given property? The semigroups S that we consider are the full transformation semigroups; all mappings from a finite set to itself under composition of mappings. The subsemigroups T that we consider are of one of the following types: left zero, right zero, completely simple, or inverse. Furthermore, we find the largest size of such subsemigroups U where the least rank of an element in U is specified. Numerous examples are given.     

On Ordered Semigroups Containing Covered Ideals

A proper ideal M of an ordered semigroup (S, ·, ≤) is said to be a covered ideal of S if M ? (S(S?M)S], i.e., if for any x in M, there exist a, c in S and b in S?M such that x ≤ abc. The purpose of this article is to study the structure of ordered semigroups containing covered ideals. The results obtained generalize the results on semigroups (without order) studied by Fabrici in 1984.     

A Notion of Rank for Right Congruences on Semigroups

ABSTRACT

We introduce a new notion of rank for a semigroup S. The rank is associated with pairs (I,ρ), where ρ is a right congruence and I is a ρ-saturated right ideal. We allow I to be the empty set; in this case the rank of (?, ρ) is the Cantor-Bendixson rank of ρ in the lattice of right congruences of S, with respect to a topology we title the finite type topology. If all pairs have rank, then we say that S is ranked. Our notion of rank is intimately connected with chain conditions: every right Noetherian semigroup is ranked, and every ranked inverse semigroup is weakly right Noetherian.

Our interest in ranked semigroups stems from the study of the class ± b? _S of existentially closed S-sets over a right coherent monoid S. It is known that for such S the set of sentences in the language of S-sets that are true in every existentially closed S-set, that is, the theory T _S of ± b? _S , has the model theoretic property of being stable. Moreover, T _S is superstable if and only if S is weakly right Noetherian. In the present article, we show that T _S satisfies the stronger property of being totally transcendental if and only if S is ranked and weakly right Noetherian.     

Semigroups with magnifiers admitting minimal subsemigroups

An element a of a semigroup S is a left magnifier if λ_a, the inner left translation associated with a, is surjective and is not injective (E. S. Ljapin [11]). When this happens there exists a proper subset M of S such that the restriction to M of λ_a is bijective. In that case M is said to be a minimal subset for the left magnifier a (F. Migliorini [13], [14], [15]). Remark that if S is a semigroup having left identities then every left magnifier of S admits minimal subsets which are right ideals. Characterisations for semigroups with left magnifiers which also contain left identities have been given by E. S. Ljapin and R. Desq, using the bicyclic monoid. The general problem, precisely to give a characterization of semigroups having left magnifiers, is still open.     

Cryptogroups with an associate subgroup

A cryptogroup is a completely regular semigroup S on which Green’s relation $\mathcal{H}$ is a congruence. For a,x∈S, x is an associate of a if a=axa. A subgroup G of S is an associate subgroup of S if it contains precisely one associate of each element of S. Further, S is a regular (respectively normal) cryptogroup if $S/\mathcal{H}$ is a regular (respectively normal) band. We provide a construction of a general (respectively regular or normal) cryptogroup in terms of groups and functions. On this model of S, we find several conditions equivalent to S containing an associate subgroup G. We characterize several varieties of completely regular semigroups, provided with the unary operation s?s ^?, where s ^? is the associate of s in G. They include completely regular semigroups, (regular, normal) cryptogroups, completely simple semigroups, and their monoid and/or overabelian members.     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.