Showing papers on "Bicyclic semigroup published in 2001"

Normal forms for free aperiodic semigroups

Abstract: The implicit operation ω is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates Using ω there is a well-defined algebra which is known as the free aperiodic semigroup In this article we introduce a specific and rather elementary list of pseudoidentitites, we show that for each n, the n-generated free aperiodic semigroup is defined by this list of pseudoidentities, and then we use this identification to show that it has a decidable word problem In the language of implicit operations, this shows that the pseudovariety of finite aperiodic semigroups is κ-recursive This completes a crucial step towards showing that the Krohn–Rhodes complexity of every finite semigroup is decidable

Free Semigroup Algebras A Survey

Abstract: This is a survey of recent results on free semigroup algebras, which are the WOT-closed algebras generated by n isometries with pairwise orthogonal ranges.

Variants of Regular Semigroups

Abstract: Let S be a regular semigroup, and let a ∈ S . Then a variant of S with respect to a is a semigroup with underlying set S and multiplication \circ defined by x \circ y = xay . In this paper, we characterise the regularity preserving elements of regular semigroups; these are the elements a such that (S,\circ) is also regular. Hickey showed that the set of regularity preserving elements can function as a replacement for the unit group when S does not have an identity. As an application, we characterise the regularity preserving elements in certain Rees matrix semigroups. We also establish connections with work of Loganathan and Chandrasekaran, and with McAlister's work on inverse transversals in locally inverse semigroups. We also investigate the structure of arbitrary variants of regular semigroups concentrating on how the local structure of a semigroup affects the structure of its variants.

On the structure of regular crypto semigroups

Abstract: 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.

On fuzzy points in semigroups

Abstract: We consider the semigroup S of the fuzzy points of a semigroup S, and discuss the relation between the fuzzy interior ideals and the subsets of S in an (intra-regular) semigroup S

Semidistributive inverse semigroups

Abstract: An inverse semigroup S is said to be meet (join) semidistributive if its lattice ( S ) of full inverse subsemigroups is meet (join) semidistributive. We show that every meet (join) semidistributive inverse semigroup is in fact distributive.

On subsemigroups of βℕ and absolute coretracts

Abstract: A semigroup S is called an absolute coretract if for any continuous homomorphism f from a compact Hausdorff right topological semigroup T onto a compact Hausdorff right topological semigroup containing S algebraically there exists a homomorphism g \colon S→ T such that f\circ g=idS. The semigroup β\ben contains isomorphic copies of any countable absolute coretract. In this article we define a class C of semigroups of idempotents each of which is a decreasing chain of rectangular semigroups. It is proved that every semigroup from C is an absolute coretract and every finite semigroup of idempotents, which is an absolute coretract, belongs to C .

Nonlinear monotone semigroups and viscosity solutions

Abstract: In a celebrated paper motivated by applications to image analysis, L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel showed that any monotone semigroup defined on the space of bounded uniformly continuous functions, which satisfies suitable regularity and locality assumptions is in fact a semigroup associated to a fully nonlinear, possibly degenerate, second-order parabolic partial differential equation. In this paper, we extend this result by weakening the assumptions required on the semigroup to obtain such a result and also by treating the case where the semigroup is defined on a general space of continuous functions like, for example, a space of continuous functions with a prescribed growth at infinity. These extensions rely on a completely different proof using in a more central way the monotonicity of the semigroup and viscosity solutions methods. Then we study the consequences on the partial differential equation of various additional assumptions on the semigroup. Finally we briefly present the adaptation of our proof to the case of two-parameters families.

The Circle Semigroup of a Ring

Abstract: Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed.

A syntactic approach to covers for E-dense semigroups over group varieties

Abstract: A major result of D. B. McAlister is that every inverse semigroup is an idempotent separating morphic image of an E -unitary inverse semigroup. The result has been generalized by various authors (including Szendrei, Takizawa, Trotter, Fountain, Almeida, Pin, Weil) to any semigroup of the following types: orthodox, regular, ii-dense with commuting idempotents, E -dense with idempotents forming a subsemigroup, and is-dense. In each case, a semigroup is a morphic image of a semigroup in which the weakly self conjugate core is unitary and separated by the homomorphism. In the present paper, for any variety H of groups and any E -dense semigroup S , the concept of an “H-verbal subsemigroup” of S is introduced which is intimately connected with the least H-congruence on S . What is more, this construction provides a short and easy access to covering results of the aforementioned kind. Moreover, the results are generalized, in that covers over arbitrary group varieties are constructed for any E -dense semigroup. If the given semigroup enjoys a “regularity condition” such as being eventually regular, group bound, or regular, then so does the cover.

Transformation semigroups and transformed dimensions

Abstract: In the transformation semigroup (X, S) we introduce the height of a closed nonempty invariant subset of X, define the transformed dimension of nonempty subset S of X and obtain some results and relations.

On smarandache algebraic structures.I: the commutative multiplicitive semigroup A(a,n)

Abstract: In this paper, under the Smarandache algorithm ,we construct a class of commutative multiplicative semigroups.

The existence of one-parameter semigroups and characterizations of operator-limit distributions

Abstract: 1. Throughout this paper we shall work with a Banach space X with the norm || • ||. We write B(X) for the algebra of continuous linear operators on X with the norm topology. By a semigroup we mean a subsemigroup of B(X) under the composition operation. In the theory of operator-limit distributions some compact semigroups associated with probability measures play a very essential role. Furthermore, it is a great importance whether there exist one-parameter semigroups in semigroups in question (cf., e.g.,[2]). The existence of one parameter semigroups was intensively investigated in the theory of compact semigroups (cf., e.g., [1] Chapter B, Section 3). The results proved there are concentrated on purely topological semigroups problems. Moreover, it seems rather complicated to apply these results in the probability on Banach spaces setting. In the paper we give a construction of one-parameter semigroups in some compact semigroups of B(X). The result could be applied directly to a problem occuring in the theory of operator-limit distributions.

On equivalence of semigroup identities

Abstract: For a given relation $\rho$ on a free semigroup ${\mathcal F}$ we describe the smallest cancellative fully invariant congruence ${\rho}^{\sharp}$ containing $\rho$. Two semigroup identities are s-equivalent if each of them is a consequence of the other on cancellative semigroups. If two semigroup identities are equivalent on groups, it is not known if they are s-equivalent. We give a positive answer to this question for all binary semigroup identities of the degree less or equal to 5. A poset of corresponding varieties of groups is given.

On Finite Semigroups Embeddable in Inverse Semigroups

Abstract: Abstract. No Abstract.

A note on certain semigroups of algebraic numbers

Abstract: The cross number κ(a) can be defined for any element a of a Krull monoid. The property κ(a) = 1 is important in the study of algebraic numbers with factorizations of distinct lengths. The arithmetic meaning of the weaker property, κ(a) ∈ Z, is still unknown, but it does define a semigroup which may be interesting in its own right. This paper studies some arithmetic (divisor theory) and analytic (distribution of elements with a given norm) properties of that semigroup and a related semigroup of ideals. 1. Notation. In the first section we consider a Krull monoidM (written multiplicatively), i.e. a commutative cancellative semigroup with a unit, for which there exists a group epimorphism v : (M) → ∐ i∈I Z of (M) (the group of quotients of M) onto a free abelian group such that M = {x ∈ (M) : vi(x) ≥ 0 for all i ∈ I}, as defined in [5] and [6]. The concept of a Krull monoid is equivalent to that of a semigroup with divisor theory, as shown in [6]. Let ∂ : M → D, where D is a free abelian semigroup, be a divisor theory for M with the class group written as Cl(M). We further assume that Cl(M) is finite and that there are infinitely many prime elements of D (prime divisors) in each class. The neutral element of Cl(M), the principal class, is denoted as H(M). If a ∈ D, then [a] will denote the class of a in the class group. In the second section we apply the results obtained for general Krull monoids to an algebraic number field K with the ring RK of algebraic integers and the semigroup I(RK) of non-zero integral ideals. We do it in the obvious way by fixing M = R K (the multiplicative semigroup) and ∂ : R K → I(RK), ∂(a) = (a). All of our previous assumptions on M are satisfied by R K for arbitrary K. In this case H denotes the class group of K, h is the class number and H stands for the class of principal ideals. The set of non-zero prime ideals of RK is written as P(RK). The Dedekind zeta-function of K is denoted by ζK . We also adopt the standard shorthand notation e(x) = exp(2πix). If X ∈ Cl(M) and a ∈ M or a ∈ D, then ΩX(a) denotes, as usual, the number of prime divisors of a in X. The cross number (cf. [4]) of elements 2000 Mathematics Subject Classification: 11R27, 11N45, 11M41.

Generalized Bicyclic Semigroups and Jones Semigroups

Abstract: In this paper, we consider the generalized bicyclic semigroups B n = 〈a, b | a n b = 1〉 and the Jones semigroups A n = 〈a, b | an+1b = a〉. They are the generalizations of the bicyclic semigroup B = 〈a, b | ab = 1〉 and its analogous semigroup A = 〈a, b | a 2 b = a〉 discovered by P.R., Jones in 1987. The word problem for these kinds of semigroups is solved. It is proved that, for n ≥ 2, B n are bisimple right inverse but not inverse semigroups and that the semigroup C = 〈a, b | a2b = a, ab2 = b〉 is the smallest idempotent-free homomorphic image of A n . Moreover, we also prove that A n and A m are mutually embeddable but not isomorphic with each other if n ≠ m. As a consequence, different kind of $${\cal D}$$ -nontrivial [0-]simple semigroups without idempotents are discussed.

An eakin-nagata theorem for semigroups

Abstract: We prove an Eakin-Nagata Theorem for commutative semigroups. The well-known Eakin-Nagata Theorem for commutative rings states the following, Theorem 1 (Eakin-Nagata). Let S be a ring, and let R be a subring of S. If S is a Noetherian ring, and if S is a nitely generated R-module, then R is also a Noetherian ring. The aim of this note is to prove an Eakin-Nagata Theorem for commutative semigroups (explicitly, for g-monoids). A submonoid of a torsion-free abelian (additive) group is called a grading monoid or a g-monoid. Let T be a g-monoid, and let S be a submonoid of T . Then T is called an extension semigroup of S, and S is called a subsemigroup of T . Let S be a g-monoid, and let M be a non-empty set so that for each pair of elements s 2 S and x 2M , there de ned s+x 2 M . If, for all s1; s2 2 S and x 2M , (s1+ s2)+x = s1 + (s2 + x) and 0 + x = x, then M is called an S-module. If there exists a nite number of elements x1; ; xn of M such that M = [ n 1 (S + xi), then M is called a nitely generated S-module. Let S be a g-monoid. If every ideal of S is nitely generated, then S is called a Noetherian semigroup. Let S be a g-monoid. If there exists a nite number of elements s1; ; sn of S such that S = P n 1 Z0si, then S is called a nitely generated g-monoid, where Z0 denotes the set of non-negative integers. If S is a nitely generated g-monoid, then S is a Noetherian semigroup. Let S be a g-monoid, and let T be an extension semigroup of S. If T is a Noetherian semigroup, and if T is a nitely generated S-module, we will prove that S is also a Noetherian semigroup. Lemma 1 ([1, Appendix]). Let R be a ring, and let S be a g-monoid. Then the semigroup ring R[X ;S] of S over R is a Noetherian ring if and only if R is a Noetherian ring and S is a nitely generated g-monoid. Theorem 1 and Lemma 1 imply the following, Proposition 1. Let S be a g-monoid, T be an extension semigroup of S, and let k be a eld. If the semigroup ring k[X ;T ] of T over k is a Noetherian ring, and if k[X ;T ] is a nitely generatef k[X ;S]-module, then S is a nitely generated g-monoid. Let S be a g-monoid, T an extension semigroup of S, and let k be a eld. Then k[X ;T ] is a nitely generated k[X ;S]-module if and only if T is a nitely generated S-module. 2000 Mathematics Subject Classi cation. Primary 13A15, Secondary 20M14.

Invariant Measures on Topological Semigroups which Have an Ideal with Open Translation Mappings

Abstract: We consider invariant measures on topological semigroups which have an ideal with open translation mappings with respect to the elements of a semigroup. We do not assume that the ideal is an open subset of a semigroup.

On one sided and naturally lattice ordered inverse semigroups

Abstract: We prove that any right amenably and naturally lattice ordered inverse semigroup is already amenably ordered. Among other things this answers a question raised in [6].