scholarly journals REALISABLE SETS OF CATENARY DEGREES OF NUMERICAL MONOIDS

2017 ◽  
Vol 97 (2) ◽  
pp. 240-245 ◽  
Author(s):  
CHRISTOPHER O’NEILL ◽  
ROBERTO PELAYO

The catenary degree is an invariant that measures the distance between factorisations of elements within an atomic monoid. In this paper, we classify which finite subsets of$\mathbb{Z}_{\geq 0}$occur as the set of catenary degrees of a numerical monoid (that is, a co-finite, additive submonoid of$\mathbb{Z}_{\geq 0}$). In particular, we show that, with one exception, every finite subset of$\mathbb{Z}_{\geq 0}$that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid.

2014 ◽  
Vol 97 (3) ◽  
pp. 289-300 ◽  
Author(s):  
SCOTT T. CHAPMAN ◽  
MARLY CORRALES ◽  
ANDREW MILLER ◽  
CHRIS MILLER ◽  
DHIR PATEL

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.


2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].


1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).


1974 ◽  
Vol 10 (1) ◽  
pp. 133-144
Author(s):  
Carlton J. Maxson
Keyword(s):  

2020 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
Gholamreza H. Mehrabani ◽  
Kourosh Nourouzi

AbstractDiversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.


Author(s):  
D. J. A. Welsh

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.


2007 ◽  
Vol 143 (6) ◽  
pp. 1493-1510 ◽  
Author(s):  
Takehiko Yasuda

AbstractFor each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.


2007 ◽  
Vol 59 (2) ◽  
pp. 343-371 ◽  
Author(s):  
Huaxin Lin

AbstractLet A be a separable amenable purely infinite simple C*-algebra which satisfies the Universal Coefficient Theorem. We prove that A is weakly semiprojective if and only if Ki(A) is a countable direct sum of finitely generated groups (i = 0, 1). Therefore, if A is such a C*-algebra, for any ε > 0 and any finite subset ℱ ⊂ A there exist δ > 0 and a finite subset ⊂ A satisfying the following: for any contractive positive linear map L : A → B (for any C*-algebra B) with ∥L(ab) – L(a)L(b)∥ < δ for a, b ∈ there exists a homomorphism h: A → B such that ∥h(a) – L(a)∥ < ε for a ∈ ℱ.


2019 ◽  
Vol 29 (03) ◽  
pp. 419-457 ◽  
Author(s):  
Alfred Geroldinger ◽  
Andreas Reinhart

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper, we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of [Formula: see text]-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions, we establish arithmetical finiteness results, in particular, for the monotone catenary degree and for the structure of sets of lengths and of their unions.


Author(s):  
P. Komjáth

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {Hi: i∈I} is a family of measurable sets with finite measure, λi (i∈I) are non-negative real numbers, then we can choose a subset Ti ⊆ Hi with μ(Ti) = λi and μ(Ti ∩ Ti′) = 0 (i ≠ i′) if and only if μ({U Hi: iεJ}) ≥ Σ{λi: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.


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