Arithmetic Congruence Monoids: A Survey

Baginski, Paul; Chapman, Scott T.

doi:10.1007/978-1-4939-1601-6_2

Cited by 12 publications

(15 citation statements)

References 20 publications

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“…For fixed n, there are hence 2 t ACMs, where t denotes the number of distinct primes dividing n, and just one of these (corresponding to u = 1) is regular. This observation was Proposition 4.1 in [2].…”

Section: Proposition 22 a Regular CM N Is Half-factorial If And Onlmentioning

confidence: 71%

“…This is quite broad in general, but a particular subclass called arithmetic congruence monoids (ACMs) have received considerable attention recently [4,[6][7][8][9][10]16]. These ACM results are surveyed in [2]. The present work considers a generalization of ACMs, still contained within the natural numbers, called congruence monoids (CMs).…”

Section: Introductionmentioning

confidence: 98%

“…For a general reference, see any of [1,2,12]. One large class of such monoids are multiplicative submonoids of the natural numbers.…”

Section: Introductionmentioning

confidence: 99%

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Arithmetic of Congruence Monoids

Fujiwara

Gibson

Jenssen

et al. 2016

Communications in Algebra

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Let N represent the positive integers. Let n ∈ N and ⊆ N. Set n = x ∈ N ∃y ∈ x ≡ y mod n ∪ 1 . If n is closed under multiplication, it is known as a congruence monoid or CM. A classical result of James and Niven [15] is that for each n, exactly one CM admits unique factorization into products of irreducibles, namely n = x ∈ N gcd x n = 1 . In this article, we examine additional factorization properties of CMs. We characterize CMs that contain primes, and we determine elasticity for several classes of CMs and bound it for several others. Also, for several classes, we characterize half-factoriality and determine whether the elasticity is accepted and whether it is full.

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Section: Proposition 22 a Regular CM N Is Half-factorial If And Onlmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 98%

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Arithmetic of Congruence Monoids

Fujiwara

Gibson

Jenssen

et al. 2016

Communications in Algebra

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“…The set of elasticities in numerical monoids was recently studied in [10]. Arithmetic congruence monoids which are not fully elastic can be found in the survey [7].…”

Section: The Set Of Elasticitiesmentioning

confidence: 99%

Sets of arithmetical invariants in transfer Krull monoids

Geroldinger

Zhong

2019

Journal of Pure and Applied Algebra

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Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between 1 and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.

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“…In Section 4, we will compute for NSCS several arithmetic properties used in factorization theory. For a general reference on factorization theory, see any of [1,2,15], and for more background on arithmetic invariants in general numerical semigroups see [5,7]. We now define the invariants considered in this paper, including the delta set (Corollary 20), catenary degree (Theorem 18) and tame degree (Theorem 21).…”

mentioning

confidence: 99%

Numerical Semigroups on Compound Sequences

Kiers

O’Neill

Ponomarenko

2016

Communications in Algebra

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We generalize the geometric sequence {a p , a p−1 b, a p−2 b 2 , . . . , b p } to allow the p copies of a (resp. b) to all be different. We call the sequence {a1a2a3 · · · ap, b1a2a3 · · · ap, b1b2a3 · · · ap, . . . , b1b2b3 · · · bp} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.

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Arithmetic Congruence Monoids: A Survey

Cited by 12 publications

References 20 publications

Arithmetic of Congruence Monoids

Arithmetic of Congruence Monoids

Sets of arithmetical invariants in transfer Krull monoids

Numerical Semigroups on Compound Sequences

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