Showing papers on "Cancellative semigroup published in 2001"

Approximation of degenerate semigroups

Abstract: By a continuous degenerate semigroup we mean a strongly continuous mapping T : R+ ! L(X) having the semigroup property. Thus, T (0) is a projection which may be different from the identity. The main theorem is a Trotter-Kato type approximation result for such degenerate semigroups. It is used to study convergence of heat semigroups with respect to variable domains.

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

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

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.

A complete theory for jointly continuous nonlinear semigroups on a complete separable metric space

Abstract: For a jointly continuous semigroup of transformations on a complete separable metric space X an induced semigroup of linear transformations on an appropriate space of measures is defined. A complete characterization of generators of such semigroups is given and it is shown how to construct a jointly continuous semigroup on X from a generator taken from this characterized collection.

Right commutative semigroups

Abstract: In this chapter we deal with semigroups which satisfy the identity axy = ayx. These semigroups are called right commutative semigroups. It is clear that a right commutative semigroup is medial and so we can use the results of the previous chapter for right commutative semigroups. For example, every right commutative semigroup is a semilattice of right commutative archimedean semigroups and is a band of right commutative t-archimedean semigroups. A semigroup is right commutative and simple if and only if it is a left abelian group. Moreover, a semigroup is right commutative and archimedean containing at least one idempotent element if and only if it is a retract extension of a left abelian group by a right commutative nil semigroup. We characterize the right commutative left cancellative and the right commutative right cancellative semigroups, respectively. Clearly, a semigroup is right commutative and left cancellative if and only if it is a commutative cancellative semigroup. A semigroup is right commutative and right cancellative if and only if it is embeddable into a left abelian group if and only if it is a left zero semigroup of commutative cancellative semigroups. It is shown that a right commutative semigroup is embeddable into a semigroup which is a union of groups if and only if it is right separative.

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.

Sum-Free Subsets of Right Cancellative Semigroups

Abstract: For a natural number k? 2 let?=?(k) be the smallest natural number which does not dividek? 1. We show that for any subset A of a right cancellative semigroup S which contains no solutions of the equation x1+? +xk=y there is an element s inS such that the setsA, A+s, . . . ,A + (?? 1)sare pairwise disjoint. In particular, if S is finite, such a set A has at most | S |/? elements. This estimate is sharp.

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.

Embedding Actions of Cancellative Directed Semigroups

Abstract: We generalize Ore's Theorem about embeddability of semigroups into groups, and obtain embeddings of semigroup actions into group actions.

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.