Showing papers in "Semigroup Forum in 2010"
87 citations
TL;DR: In this article, the authors introduce three kinds of resolvent families defined by purely algebraic equations, which extend the classical semigroup property and Cosine functional equation, and give their basic properties and analyticity criteria.
Abstract: In this paper we introduce three kinds of resolvent families defined by purely algebraic equations, which extend the classical semigroup property and Cosine functional equation. We give their basic properties and analyticity criteria. Moreover, the relations between integrated resolvent families and resolvent families are discussed as well.
71 citations
TL;DR: In this paper, it was shown that the monoid of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity (a.k.a., A^2B^4A^2A^A^B+A+B+C+C^A+C +B+B +A^C+A +C+B^C +A+2B+4A+A−2B +C^C^2C+4B+2A−4A +2B−4B +4A
Abstract: We show that the monoid $M_{2}(\mathbb {T})$
of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity $$A^2B^4A^2A^2B^2A^2B^4A^2=A^2B^4A^2B^2A^2A^2B^4A^2.$$
Studying reduced identities for subsemigroups of $M_{2}(\mathbb {T})$
, and introducing a faithful semigroup representation for the bicyclic monoid by 2×2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.
45 citations
TL;DR: In this paper, a simpler approach to counting numerical semigroups of a given genus was proposed, without referring to the generators or the semigroup tree, and an improved asymptotic lower bound was given.
Abstract: Let ng denote the number of numerical semigroups of genus g. Bras-Amoros conjectured that ng possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree. We offer a new, simpler approach to counting numerical semigroups of a given genus. Our method gives direct constructions of families of numerical semigroups, without referring to the generators or the semigroup tree. In particular, we give an improved asymptotic lower bound for ng.
42 citations
TL;DR: In this article, the authors defined the module topological center of the second dual of a Banach algebra and proved that it is amenable if and only if an appropriate group homomorphic image of S is finite.
Abstract: In this paper we define the module topological center of the second dual $\mathcal{A}^{**}$
of a Banach algebra $\mathcal{A}$
which is a Banach $\mathfrak{A}$
-module with compatible actions on another Banach algebra $\mathfrak{A}$
. We calculate the module topological center of l
1(S)**, as an l
1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that l
1(S)** is l
1(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.
41 citations
TL;DR: In this paper, the authors investigated the algebraic structure of this approach and dispense with the restriction to finitely generated monoids and give applications to other invariants of non-unique factorizations, such as the elasticity and the set of distances.
Abstract: The investigation and classification of non-unique factorization phenomena have attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P. Garcia-Sanchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations, and they applied this method successfully in the case of numerical monoids. In this paper, we investigate the algebraic structure of this approach. Thereby, we dispense with the restriction to finitely generated monoids and give applications to other invariants of non-unique factorizations, such as the elasticity and the set of distances.
35 citations
TL;DR: In this paper, the role of the module topological center in module amenability of closed subalgebras of a Banach algebra was investigated. And the authors showed that the topological centre plays an important role in module amenableness.
Abstract: For a Banach algebra $\mathcal{A}$
which is also an $\mathfrak{A}$
-bimodule, we study relations between module amenability of closed subalgebras of $\mathcal{A}''$
, which contains $\mathcal{A}$
, and module Arens regularity of $\mathcal{A}$
and the role of the module topological centre in module amenability of $\mathcal{A}''$
. Then we apply these results to $\mathcal{A}=l^{1}(S)$
and $\mathfrak{A}=l^{1}(E)$
for an inverse semigroup S with subsemigroup E of idempotents. We also show that l
1(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group $\frac{S}{\approx}$
, is amenable. Moreover, we define super module amenability and show that l
1(S) is super module amenable if and only if $\frac{S}{\approx}$
is finite.
27 citations
TL;DR: In this paper, the notion of injectivity in Pos-S of S-posets for a pomonoid S is studied and a homological classification of pomonoids and pogroups is given.
Abstract: In this paper we study the notion of injectivity in the category Pos-S of S-posets for a pomonoid S. First we see that, although there is no non-trivial injective S-poset with respect to monomorphisms, Pos-S has enough (regular) injectives with respect to regular monomorphisms (sub S-posets). Then, recalling Banaschewski’s theorem which states that regular injectivity of posets with respect to order-embeddings and completeness are equivalent, we study regular injectivity for S-posets and get some homological classification of pomonoids and pogroups. Among other things, we also see that regular injective S-posets are exactly the retracts of cofree S-posets over complete posets.
25 citations
TL;DR: In this article, the undamped linear wave equation on a bounded domain in ℝn with C2 boundary is considered, and the model is reformulated as an abstract first order Cauchy problem in an appropriate Hilbert space, including the memory of the boundary as a state component.
Abstract: The undamped linear wave equation on a bounded domain in ℝn with C2 boundary is considered. The interaction of the interior waves and the viscoelastic boundary material is modeled by convolution boundary conditions. It is assumed that the convolution kernel is integrable and completely monotonic. The main result is that the derivatives of all solutions tend to zero. The proof is given by an application of the Arendt-Batty-Lyubic-Vu Theorem. To this end, the model is reformulated as an abstract first order Cauchy problem in an appropriate Hilbert space, including the memory of the boundary as a state component. It is shown that the differential operator of the Cauchy problem is the generator of a contraction semigroup on the state space by establishing the range condition for the Lumer-Phillips Theorem using a generalized Lax-Milgram argument and Fredholm’s alternative. Furthermore, it is shown that neither the generator nor its adjoint have purely imaginary eigenvalues.
22 citations
TL;DR: In this article, the existence of mild solutions for a class of impulsive neutral functional differential equations with state-dependent delay is established, and mild solutions have been shown to be possible for some classes of neutral functions.
Abstract: In this work, we establish the existence of mild solutions for a class of impulsive neutral functional differential equations with state-dependent delay.
20 citations
TL;DR: In this paper, the authors describe a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal'cev products with the pseudovarsieties of definite and reverse definite semigroups.
Abstract: The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal’cev products with the pseudovarieties of definite and reverse definite semigroups.
TL;DR: In this article, the authors studied some properties of semigroups with presentation aOE (c) a,b ; a (p) =b (r),a (q) = b (s) > and studied their potential as platforms for the Diffie-Hellman key exchange protocol.
Abstract: In this paper, we study some properties of semigroups with presentation aOE (c) a,b ; a (p) =b (r) ,a (q) =b (s) > We will also study their potential as platforms for the Diffie-Hellman key exchange protocol
TL;DR: In this article, the pseudo almost automorphic solution of a fractional order neutral differential equation in a Banach space X is discussed. The results are established using the Krasnoselskii's fixed point theorem.
Abstract: In this paper we discuss the pseudo almost automorphic solution of a fractional order neutral differential equation in a Banach space X. The results are established using the Krasnoselskii’s fixed point theorem.
TL;DR: In this paper, it was shown that if a semigroup T divides into a semidirect product S⋊T where S is a finite semilattice whose natural order makes S a chain, then so does any semi-directional product S ⋊ T where T is a chain.
Abstract: We show that if a semigroup T divides a semigroup of full order preserving transformations of a finite chain, then so does any semidirect product S⋊T where S is a finite semilattice whose natural order makes S a chain.
TL;DR: In this article, it was shown that Thompson's group F is the symmetry group of the generic idempotent, i.e., the group of automorphisms of an object A and an isomorphism A⊗A→A.
Abstract: We show that Thompson’s group F is the symmetry group of the ‘generic idempotent’. That is, take the monoidal category freely generated by an object A and an isomorphism A⊗A→A; then F is the group of automorphisms of A.
TL;DR: In this article, Borichev and Tomilov show that the type of the generated semigroup is determined by a matrix, originating from the domain of the operator, and provide necessary and sufficient conditions for exponential and polynomial stability of the semigroup in terms of the matrix.
Abstract: In this paper we examine difference operators with constant coefficients. We show that the type of the generated semigroup is determined by a matrix \(\mathbb{B}\), originating from the domain of the operator. Moreover, we provide necessary and sufficient conditions for exponential and polynomial stability of the semigroup in terms of the matrix \(\mathbb{B}\), using results of A. Borichev and Y. Tomilov. We close the paper with an application of our results to flows in networks.
TL;DR: In this article, the authors studied the joint weak continuity of multiplication on LUC(G) and how the joint continuity property of LUC can be related to a property concerning the whole algebra.
Abstract: Let G be a locally compact group and LUC(G) the C*-algebra of the bounded left uniformly continuous functions on G. The spectrum G
LUC of LUC(G) is the universal semigroup compactification of G with respect to the joint continuity property: the multiplication on G×G
LUC is jointly continuous. The paper studies the joint weak* continuity of multiplication on LUC(G)* and, in particular, the question how the joint continuity property of G
LUC can be related to a property concerning the whole algebra LUC(G)*. The group G is naturally replaced by the measure algebra M(G), and LUC(G)* can be identified with M(G
LUC), the space of regular Borel measures on G
LUC. It is shown that the joint weak* continuity can fail even on bounded sets of M(G)×M(G
LUC), but, on the other hand, the multiplication on M(G)×M(G
LUC) is positive continuous in the sense of Jewett.
TL;DR: In this article, the least distributive lattice congruence on the semirings in the variety of all semiring whose additive reduct is a semilattice is described.
Abstract: We describe the least distributive lattice congruence on the semirings in the variety of all semirings whose additive reduct is a semilattice, introduce the notion of a k-Archimedean semiring and characterize the semirings that are distributive lattices or chains of k-Archimedean semirings.
TL;DR: In this paper, the authors characterize the endomorphisms of the semigroup of all order-preserving mappings on a finite chain and show that there are three types of endomorphism: automorphisms, constants, and a certain type with two idempotents in the image.
Abstract: We characterize the endomorphisms of the semigroup of all order-preserving mappings on a finite chain. We show that there are three types of endomorphism: automorphisms, constants, and a certain type of endomorphism with two idempotents in the image.
TL;DR: In this article, the authors give a structure theorem for proper left restriction semigroups and show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid.
Abstract: Proper extensions that are “injective on ℒ-related idempotents” of ℛ-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a λ-semidirect product of a left regular band by an ℛ-unipotent or by a glrac semigroup, respectively An example of such is the generalized Szendrei expansion As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semilattice
TL;DR: In this paper, the question of whether a finite involution semigroup is inherently non-finitely based (INFB) was investigated, which means that it is not contained in any finitely based locally finite variety.
Abstract: We investigate the question of whether a finite involution semigroup is inherently nonfinitely based (INFB), which means that it is not contained in any finitely based locally finite variety. Although we fall short of a full characterization, we nevertheless clarify a number of interesting subcases.
TL;DR: A pomonoidS is a monoid equipped with a partial order that is compatible with the binary operation as discussed by the authors, where S-posets correspond to the representation of S by order preserving transformations of posets.
Abstract: A pomonoidS is a monoid equipped with a partial order that is compatible with the binary operation. In the same way that M-acts over a monoidM correspond to the representation of M by transformations of sets, S-posets correspond to the representation of a pomonoid S by order preserving transformations of posets.
TL;DR: In this article, the Cayley D-saturated property of a semigroup S with respect to any subset T⊆S is characterized, and a subgraph of Cay(S,T) isomorphic to D with all vertices in V.
Abstract: Let D be a finite graph. A semigroup S is said to be Cayley D-saturated with respect to a subset T of S if, for all infinite subsets V of S, there exists a subgraph of Cay(S,T) isomorphic to D with all vertices in V. The purpose of this paper is to characterize the Cayley D-saturated property of a semigroup S with respect to any subset T⊆S. In particular, the Cayley D-saturated property of a semigroup S with respect to any subsemigroup T is characterized.
TL;DR: Factorizable inverse monoids constitute the algebraic theory of partial symmetries which are restrictions of automorphisms; the formal definition is that each element is the product of an idempotent and an invertible as discussed by the authors.
Abstract: Factorizable inverse monoids constitute the algebraic theory of those partial symmetries which are restrictions of automorphisms; the formal definition is that each element is the product of an idempotent and an invertible. This class of monoids has theoretical significance, and includes concrete instances which are important in various contexts. This survey is organised around the idea of group acts on semilattices and contains a large range of examples. Topics also include methods for construction of factorizable inverse monoids, and aspects of their inner structure, morphisms, and presentations.
TL;DR: In this article, it was shown that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L1(Ω) under a duality method, and estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator.
Abstract: Given an open domain (possibly unbounded) Ω⊂Rn, we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L1(Ω) We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order
TL;DR: In this article, the singular part of the transformation semigroup is studied in terms of generators and relations, where the symmetric group is defined as the group of all permutations on the finite set {1,…,n}.
Abstract: The (full) transformation semigroup \(\mathcal{T}_{n}\) is the semigroup of all functions from the finite set {1,…,n} to itself, under the operation of composition. The symmetric group \({\mathcal{S}_{n}\subseteq \mathcal{T}_{n}}\) is the group of all permutations on {1,…,n} and is the group of units of \(\mathcal{T}_{n}\). The complement \(\mathcal{T}_{n}\setminus \mathcal{S}_{n}\) is a subsemigroup (indeed an ideal) of \(\mathcal{T}_{n}\). In this article we give a presentation, in terms of generators and relations, for \(\mathcal{T}_{n}\setminus \mathcal{S}_{n}\), the so-called singular part of \(\mathcal{T}_{n}\).
TL;DR: In this paper, a more direct semigroup-theoretic treatment of abstract delay differential equations, a new derivation of the form of the McKendrick semigroup, and a generation theorem for a semigroup describing kinase activity were presented.
Abstract: We show that Lord Kelvin’s method of images is a way to prove generation theorems for semigroups of operators. To this end we exhibit three examples: a more direct semigroup-theoretic treatment of abstract delay differential equations, a new derivation of the form of the McKendrick semigroup, and a generation theorem for a semigroup describing kinase activity in the recent model of Kaźmierczak and Lipniacki (J. Theor. Biol. 259:291–296, 2009).
TL;DR: In this article, the set X under consideration is a totally ordered set with n points, and the set of all order-preserving transformations in X form a regular semigroup.
Abstract: Let TX be the full transformation semigroup on a set X,
$$T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}$$
be the subsemigroup of TX determined by an equivalence E on X. In this paper the set X under consideration is a totally ordered set with n points. The set of all order preserving transformations in \(T_{E^{*}}(X)\) forms a subsemigroup of \(T_{E^{*}}(X)\) denoted by
$$O_{E^*}(X)=\{\alpha\in T_{E^*}(X): \forall x,y\in X, x\leq y\Rightarrow x\alpha\leq y\alpha\}.$$
In this paper, we discuss Green’s relations for \(O_{E^{*}}(X)\) and prove that \(O_{E^{*}}(X)\) is a regular semigroup.
TL;DR: In this paper, a class of inverse monoids of the form M =Inv √ X √ w = 1 is studied, where the relator w has a combinatorial property that is called sparse.
Abstract: We study a class of inverse monoids of the form M=Inv 〈X∣w=1〉, where the single relator w has a combinatorial property that we call sparse. For a sparse word w, we prove that the word problem for M is decidable. We also show that the set of words in (X∪X
−1)* that represent the identity in M is a deterministic context free language, and that the set of geodesics in the Schutzenberger graph of the identity of M is a regular language.
TL;DR: In this paper, the E-order-preserving transformation semigroup on a totally ordered set (X,≤) is defined and a subsemigroup of X is investigated.
Abstract: Let (X,≤) be a totally ordered set, T
X
the full transformation semigroup on X and E an arbitrary equivalence on X. We consider a subsemigroup of T
X
defined by $$\mathit{EOP}_X=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E~\hbox{and}~x\leq y\Rightarrow(x\alpha,y\alpha)\in E~\hbox{and}~x\alpha\leq y\alpha\}$$
and call it the E-order-preserving transformation semigroup on
X. We investigate regularity and Green’s relations of EOP
X
when X is finite.