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Showing papers on "Cancellative semigroup published in 2013"


Journal ArticleDOI
Hirokatsu Nari1
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


Journal ArticleDOI
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


Journal ArticleDOI
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


Journal ArticleDOI
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


Journal ArticleDOI
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


Journal ArticleDOI
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


Journal ArticleDOI
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


Posted Content
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


Journal ArticleDOI
Evan O'Dorney1
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.

12 citations


Journal ArticleDOI
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.

11 citations


Journal ArticleDOI
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.

Journal ArticleDOI
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.

Journal ArticleDOI
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.

Journal ArticleDOI
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.

Journal ArticleDOI
TL;DR: It is shown that being finitely presentable and being finally presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids.
Abstract: We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterization of finite presentability for left cancellative monoids. We also give examples to show that this characterization does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.

Journal ArticleDOI
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.

Journal Article
TL;DR: In this article, the structure of cancellative quasi-commutative primary ternary semigroups was studied and it was shown that the proper prime ideals in T are maximal and the semiprimary ideals are equivalent.
Abstract: I n this paper we study the structure of cancellative quasi-commutative primary ternary semigroups. In fact we prove that if T is a cancellative quasi-commutative ternary semigroup, then (1) S is a primary ternary semigroup (2) proper prime ideals in T are maximal and (3) semiprimary ideals in T are primary, are equivalent.

Journal ArticleDOI
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.

Journal ArticleDOI
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.

Journal ArticleDOI
TL;DR: For an arbitrary set X (finite or infinite), the symmetric inverse semigroup of partial injective transformations on X is defined in this article, and the structure of C(a) is determined in terms of Green's relations.
Abstract: For an arbitrary set X (finite or infinite), denote by I(X) the symmetric inverse semigroup of partial injective transformations on X. For an element a in I(X), let C(a) be the centralizer of a in I(X). For an arbitrary a in I(X), we characterize the elements b in I(X) that belong to C(a), describe the regular elements of C(a), and establish when C(a) is an inverse semigroup and when it is a completely regular semigroup. In the case when the domain of a is X, we determine the structure of C(a) in terms of Green's relations. 10.1017/S0004972712000779

Journal ArticleDOI
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.

Journal ArticleDOI
TL;DR: In this article, the authors give conditions which are both necessary and sufficient for a subsemigroup T to be left amenable to a cancellative, left-amenable semigroup S.
Abstract: Let T be a subsemigroup of a cancellative, left amenable semigroup S. We give conditions which are both necessary and sufficient for the subsemigroup T to be left amenable.

Journal ArticleDOI
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.

Journal ArticleDOI
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.

Posted Content
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).

Journal ArticleDOI
TL;DR: In this paper, a class of irreducible non-inverse representations of the semigroup ℤ+\{1] is introduced and a complete description of such representations of that semigroup is provided.
Abstract: In this paper we study isometric representations of the semigroup ℤ+\{1}. The notion of inverse representation is introduced and a complete (to within unitary equivalence) description of such representations of that semigroup is provided. A class of irreducible non-inverse representations (β-representations of the semigroup ℤ+\{1}) is described.

Journal ArticleDOI
TL;DR: In this article, it was shown that if G is a countable discrete group, p is a right cancelable element of G∗ = βG\\G, and λ is an ordinal, then there is a strictly decreasing chain of idempotents in Cp, the smallest compact subsemigroup of G ∗ with p as a member.
Abstract: Given idempotents e and f in a semigroup, e ≤ f if and only if e = fe = ef . We show that if G is a countable discrete group, p is a right cancelable element of G∗ = βG\\G, and λ is a countable ordinal, then there is a strictly decreasing chain 〈qσ〉σ<λ of idempotents in Cp, the smallest compact subsemigroup of G∗ with p as a member. We also show that if S is any infinite subsemigroup of a countable group, then any nonminimal idempotent in S∗ is the largest element of such a strictly decreasing chain of idempotents. (It had been an open question as to whether there was a strictly decreasing chain 〈qσ〉σ<ω+1 in N∗.) As other corollaries we show that if S is an infinite right cancellative and weakly left cancellative discrete semigroup, then βS contains a decreasing chain of idempotents of reverse order type λ for every countable ordinal λ and that if S is an infinite cancellative semigroup then the set U(S) of uniform ultrafilters contains such decreasing chains.

Posted Content
TL;DR: In this article, it was shown that the uniform word problem for word-hyperbolic semigroups is solvable in polynomial time (improving on the previous exponential-time algorithm).
Abstract: This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.

Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient criterion for a numerical semigroup to be a Weierstras semigroup is given, and the space of nodal curves with Weierstrauss semigroups is studied.
Abstract: We continue the investigation of curves of type p, q started in Knebl et al. (J Algebra 348:315–335, 2011). We study the space of such curves and the space of nodal curves with prescribed Weierstras semigroup. A necessary and sufficient criterion for a numerical semigroup to be a Weierstras semigroup is given. Using this criterion we find a class of Weierstras semigroups which apparently has not yet been described in the literature.