Showing papers on "Bicyclic semigroup published in 2013"
TL;DR: Barucci and Froberg as discussed by the authors characterized almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers and gave a criterion for H� ∗ (the dual of M) to be an almost-symmetric semigroup.
Abstract: The notion of an almost symmetric numerical semigroup was given by V. Barucci and R. Froberg in J. Algebra, 188, 418–442 (1997). We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for H
∗ (the dual of M) to be an almost symmetric numerical semigroup. Using these results we give a formula for the multiplicity of an opened modular numerical semigroup. Finally, we show that if H
1 or H
2 is not symmetric, then the gluing of H
1 and H
2 is not almost symmetric.
55 citations
TL;DR: In this article, the authors present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semiigroup S and a semigroup ideal E⊆S, produces a new numerical semi-givers, denoted by S⋈bE (where b is any odd integer belonging to S), such that S=(S⋆bE)/2.
Abstract: In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup S and a semigroup ideal E⊆S, produces a new numerical semigroup, denoted by S⋈bE (where b is any odd integer belonging to S), such that S=(S⋈bE)/2. In particular, we characterize the ideals E such that S⋈bE is almost symmetric and we determine its type.
27 citations
TL;DR: Two new types of hereditarily finitely based identity systems are presented and two of these systems establish the hereditary finite basis property of every semigroup of order five or less with one possible exception.
Abstract: A system of semigroup identities is hereditarily finitely based if it defines a variety all semigroups of which are finitely based. Two new types of hereditarily finitely based identity systems are presented. Two of these systems, together with eight existing systems, establish the hereditary finite basis property of every semigroup of order five or less with one possible exception.
25 citations
TL;DR: In this paper, the proportion of numerical semigroups which do not satisfy the Buchweitz criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve was studied.
21 citations
TL;DR: In this paper, it was shown that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup S and all group congruences on S, where κ is the least E-unitary congruence.
Abstract: We show that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup S and the set of all group congruences on S. We describe also group congruences on E-inversive (E-)semigroups. In particular, we generalize the result of Meakin (J. Aust. Math. Soc. 13:259–266, 1972) concerning the description of the least group congruence on an orthodox semigroup, the result of Howie (Proc. Edinb. Math. Soc. 14:71–79, 1964) concerning the description of ρ∨σ in an inverse semigroup S, where ρ is a congruence and σ is the least group congruence on S, some results of Jones (Semigroup Forum 30:1–16, 1984) and some results contained in the book of Petrich (Inverse Semigroups, 1984). Also, one of the main aims of this paper is to study of group congruences on E-unitary semigroups. In particular, we prove that in any E-inversive semigroup, \(\mathcal{H}\cap\sigma\subseteq\kappa\), where κ is the least E-unitary congruence. This result is equivalent to the statement that in an arbitrary E-unitary E-inversive semigroup S, \(\mathcal{H}\cap\sigma= 1_{S}\).
19 citations
TL;DR: In this paper, the authors adapted the notion of almost left factorizability to restriction semigroups and proved that each restriction semigroup is embeddable into an almost left-factorizable restriction semgroup.
Abstract: The notion of almost left factorizability and the results on almost left factorizable weakly ample semigroups, due to Gomes and the author, are adapted for restriction semigroups. The main result of the paper is that each restriction semigroup is embeddable into an almost left factorizable restriction semigroup. This generalizes a fundamental result of the structure theory of inverse semigroups.
18 citations
TL;DR: Given an arbitrary group G, it is shown that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G, and if G is finitely presented then BG is finite.
Abstract: Given an arbitrary group $G$ we construct a semigroup of idempotents (band) $B_G$ with the property that the free idempotent generated semigroup over $B_G$ has a maximal subgroup isomorphic to $G$. If $G$ is finitely presented then $B_G$ is finite. This answers several questions from recent papers in the area.
18 citations
TL;DR: In this article, the subsemigroup T(X,Y) of the full transformation semigroup on a set X and a nonempty Y⊆X is shown to be right abundant but not left abundant whenever Y is a proper non-singleton subset of X.
Abstract: Given a set X and a nonempty Y⊆X, we denote by T(X,Y) the subsemigroup of the full transformation semigroup on X consisting of all transformations whose range is contained in Y. We show that the semigroup T(X,Y) is right abundant but not left abundant whenever Y is a proper non-singleton subset of X.
13 citations
TL;DR: This work has implemented the Delorme algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical Semigroups, and numerical semIGroups associated to an irreducible plane curve singularity.
Abstract: Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to an irreducible plane curve singularity. The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families.
13 citations
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TL;DR: In this article, the authors studied self-adjoint semigroups of partial isometries on a Hilbert space, and obtained a general structure result that every semigroup consists of "generalized weighted composition" operators on a space of square-integrable Hilbert space valued functions.
Abstract: We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of "generalized weighted composition" operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is irreducible and contains a compact operator then the underlying measure space is purely atomic, so that the semigroup is represented as "zero-unitary" matrices. In this case it is not even required that the semigroup be self-adjoint.
12 citations
TL;DR: In this paper, the authors investigated the problem of estimating the number of semigroups at depth g with h children, showing that as g becomes large, it tends to a proportion φ −h−2 of all numerical semigroup, where φ is the golden ratio.
Abstract: A numerical semigroup is a subset Λ of the nonnegative integers that is closed under addition, contains 0, and omits only finitely many nonnegative integers (called the gaps of Λ). The collection of all numerical semigroups may be visually represented by a tree of element removals, in which the children of a semigroup Λ are formed by removing one element of Λ that exceeds all existing gaps of Λ. In general, a semigroup may have many children or none at all, making it difficult to understand the number of semigroups at a given depth on the tree. We investigate the problem of estimating the number of semigroups at depth g with h children, showing that as g becomes large, it tends to a proportion φ
−h−2 of all numerical semigroups, where φ is the golden ratio.
TL;DR: In this article, the authors describe all interassociates of a free commutative semigroup and prove that each interassociate is a variant of it or coincides with that semigroup.
Abstract: We describe all interassociates of a free commutative semigroup. Each interassociate of a free commutative semigroup is proved to be a variant of it or coincides with that semigroup. Conditions are found for the isomorphy of two variants of a free commutative semigroup.
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TL;DR: In this paper, Li's construction of semigroup $C^*$-algebras was used to associate a semigroup with Zappa-Szep products and give an explicit presentation of the algebra.
Abstract: Zappa-Szep products of semigroups encompass both the self-similar group actions of Nekrashevych and the quasi-lattice-ordered groups of Nica. We use Li's construction of semigroup $C^*$-algebras to associate a $C^*$-algebra to Zappa-Szep products and give an explicit presentation of the algebra. We then define a quotient $C^*$-algebra that generalises the Cuntz-Pimsner algebras for self-similar actions. We indicate how known examples, previously viewed as distinct classes, fit into our unifying framework. We specifically discuss the Baumslag-Solitar groups, the binary adding machine, the semigroup $\mathbb{N}\rtimes\mathbb{N}^\times$, and the $ax+b$-semigroup $\mathbb{Z}\rtimes\mathbb{Z}^\times$.
TL;DR: In this paper, the minimal number of matrices on Cn required to form a hypercyclic abelian semigroup is shown to be n+1, and it is shown that the action of any ABELIAN semigroup finitely generated by matrix on Cnor Rn is not k-transitive for k 2.
Abstract: We prove that the minimal number of matrices on Cn required to forma hypercyclic abelian semigroup on Cn is n+1. We also prove that theaction of any abelian semigroup finitely generated by matrices on Cnor Rn is never k-transitive for k 2. These answer questions raised byFeldman and Javaheri.
TL;DR: It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based, therefore for finite restrictionSemigroups, the existence of a finite basis is decidable "modulo monoids".
Abstract: The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups Regarded in that fashion, they have long been known to be finitely based The semigroup B2 carries the natural structure of an inverse semigroup Regarded as such, in the signature {⋅, -1}, it is also finitely based It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based) The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2 It is again inherently nonfinitely based, regarded in that fashion It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids" These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻 For example, explicit bases of identities are found for the varieties generated by B0 and B2
TL;DR: In this article, the authors give a full description of right units of the semigroup BX(D), which are defined by semilattices of the class net, for the case where X is a finite set.
Abstract: In this paper, we give a full description of right units of the semigroup BX(D), which are defined by semilattices of the class net. For the case where X is a finite set, we derive formulas by calculating the numbers of right units of the corresponding semigroup.
TL;DR: In this paper, the approximate amenability of semigroup algebra l 1 (S ) is investigated, where (S) is a uniformly locally finite inverse semigroup and S is a band semigroup.
TL;DR: In this article, it was shown that the special semigroup amalgam with core as a quasi-unitary subsemigroup is embedded in a semigroup, and that it is also embeddable in a left [right] quasinormal band.
Abstract: We show that the special semigroup amalgam with core as a quasi-unitary subsemigroup is embedded in a semigroup. We also show that the special semigroup amalgam within the class of left [right] quasinormal bands is embeddable in a left [right] quasinormal band.
TL;DR: In this paper, it was shown that such a nice explicit representation of the Dirichlet-to-Neumann semigroup is not possible for any domain except Euclidean balls, and they presented a natural and probably the simplest numerical scheme to calculate this semigroup in full generality by using Chernoff's theorem.
Abstract: In his book (Functional Analysis, Wiley, New York, 2002), P. Lax constructs an explicit representation of the Dirichlet-to-Neumann semigroup, when the matrix of electrical conductivity is the identity matrix and the domain of the problem in question is the unit ball in ℝn. We investigate some representations of Dirichlet-to-Neumann semigroup for a bounded domain. We show that such a nice explicit representation as in Lax book, is not possible for any domain except Euclidean balls. It is interesting that the treatment in dimension 2 is completely different than other dimensions. Finally, we present a natural and probably the simplest numerical scheme to calculate this semigroup in full generality by using Chernoff’s theorem.
TL;DR: In this article, the amenability of the semigroup algebra l1(S, ω) and its second dual for an arbitrary semigroup S is investigated, where S is a regular Rees matrix semigroup.
Abstract: In this work, we will describe the weighted semigroup algebra l1(S, ω), where S is a regular Rees matrix semigroup and ω ≥ 1. Then as an application, we investigate the amenability of the semigroup algebra l1(S, ω) and its second dual for an arbitrary semigroup S.
TL;DR: In this paper, the authors consider several classes of complete intersection numerical semigroups from many different contexts like algebraic geometry, commutative algebra, coding theory and factorization theory.
Abstract: We consider several classes of complete intersection numerical semigroups, aris- ing from many different contexts like algebraic geometry, commutative algebra, coding theory and factorization theory. In particular, we determine all the logical implications among these classes and provide examples. Most of these classes are shown to be well-behaved with respect to the operation of gluing.
TL;DR: In this paper, isometric representations of the semigroup ℤ+\{1] have been studied and the notion of an inverse representation is introduced and a complete description of such representations is given up to unitary equivalence.
Abstract: We study isometric representations of the semigroup ℤ+\{1}. Notion of an inverse representation is introduced and a complete description (up to unitary equivalence) of such representations is given. Also, we study a class of non-inverse irreducible representations — β-representations of the semigroup ℤ+\{1}.
TL;DR: In this paper, a geometric fractal-like structure on the affine space was proposed for solving the two-dimensional Jacobian Conjecture, which is much more general than the structure suggested by Kambayashi for etale polynomial mappings in the algebraic context.
Abstract: The motivation for this paper comes from new ideas for solving the two-dimensional Jacobian Conjecture. The Jacobian Conjecture is one of the most famous open problems in algebraic geometry. This long-standing conjecture is no doubt one of the central problems in this well developed field of mathematics and hence the importance of investigating it. We can consider a semigroup of local diffeomorphisms on the affine space with a composition of mappings as its binary operation. We put a geometric fractal-like structure on this semigroup after equipping it with a natural metric (this is heavily dependent on the fact that our mappings are local diffeomorphisms). This structure is much more general than the structure of the ind-variety suggested by Kambayashi for etale polynomial mappings in the algebraic context. Hence, it applies to other semigroups such as the semigroup of all the entire functions in one complex variable with a nonvanishing first order derivative. This last semigroup is the theme of the current paper. We hope that the corresponding Hausdorff measure and Hausdorff dimension will enable us to relate the structure of the semigroup with arithmetic machinery such as certain Zeta functions.
TL;DR: In this paper, the broken Dirichlet convolution and the broken bicyclic semigroup were introduced by the same elementary categorical construction, and both have a common origin and are introduced using a compatible partition of a monoid.
Abstract: The paper deals with certain breaking processes using a compatible partition of a monoid. We introduce the broken Dirichlet convolution and the broken bicyclic semigroup. Both have a common origin and are introduced by the same elementary categorical construction.
TL;DR: In this paper, the authors investigated the Mobius function associated to (ℤ,≤¯¯¯¯ S petertodd ) when S is an arithmetic semigroup and showed that it is locally finite.
Abstract: Let S be a numerical semigroup and let (ℤ,≤
S
) be the (locally finite) poset induced by S on the set of integers ℤ defined by x≤
S
y if and only if y−x∈S for all integers x and y. In this paper, we investigate the Mobius function associated to (ℤ,≤
S
) when S is an arithmetic semigroup.
TL;DR: In this article, the natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup.
Abstract: Mitsch's natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup. The corresponding sublattice for the partition monoid is also described.
TL;DR: In this article, the authors considered η∗-simple semigroups with zero which contain a least non-zero idempotent semigroup and showed that η-simple with zero is a semigroup with no semilattice congruences.
Abstract: A semigroup S is called η-simple if S has no semilattice congruences except S×S. Tamura in (Semigroup Forum 24:77–82, 1982) studied η-simple semigroups with a unique idempotent. In the present paper we consider a more general situation, that is, we investigate η-simple semigroups (without zero) with a least idempotent. Moreover, we study η∗-simple semigroups with zero which contain a least non-zero idempotent.
TL;DR: In this paper, the authors show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
Abstract: Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
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TL;DR: In this paper, a semigroup is called an equational domain if any finite union of algebraic sets over the set is algebraic and any semigroup with nontrivial center is not equational.
Abstract: A semigroup $S$ is called an equational domain if any finite union of algebraic sets over $S$ is algebraic. For a finite simple semigroup we find necessary and sufficient conditions to be an equational domain. Moreover, we study semigroups with nontrivial center and prove that any such semigroup is not an equational domain.
Journal Article•
TL;DR: In this paper, the authors consider biatness of certain classes of band semigroup algebras and give a necessary condition for a band semi-group algebra to be biat and show that this condition is not sufficient.
Abstract: In the present paper, we consider biatness of certain classes of semigroup algebras. Indeed, we give a necessary condi- tion for a band semigroup algebra to be biat and show that this condition is not sucient. Also, for a certain class of inverse semi- groups S; we show that the biatness of ' 1 (S) 00 is equivalent to the biprojectivity of ' 1 (S).