Catenary Degree Research Articles

Factorization in additive monoids of evaluation polynomial semirings

For a positive real α, we can consider the additive submonoid M of the real line that is generated by the nonnegative powers of α. When α is transcendental, M is a unique factorization monoid. However, when α is algebraic, M may not be atomic, and even when M is atomic, it may contain elements having more than one factorization (i.e., decomposition as a sum of irreducibles). The main purpose of this paper is to study the phenomenon of multiple factorizations inside M. When α is algebraic but not rational, the arithmetic of factorizations in M is highly interesting and complex. In order to arrive to that conclusion, we investigate various factorization invariants of M, including the sets of lengths, sets of Betti elements, and catenary degrees. Our investigation gives continuity to recent studies carried out by Chapman et al. in 2020 and by Correa-Morris and Gotti in 2022.

Gömme Boyutu Üç Olan Bazı Pseudo-Simetrik Sayısal Yarıgruplarının Delta Kümeleri Üzerine

Let S be a numerical semigroup. The catenary degree of an element s in S is a non-negative integer used to measure the distance between factorizations of s. The catenary degree of the numerical semigroup S is obtained at the maximum catenary degree of its elements. The maximum catenary degree of S is attained via Betti elements of S with complex properties. The Betti elements of S can be obtained from all minimal presentations of S. A presentation for S is a system of generators of the kernel congruence of the special factorization homomorphism. A presentation is minimal if it can not be converted to another presentation, that is, any of its proper subsets is no longer a presentation. The Delta set of S is a factorization invariant measuring the complexity of sets of the factorization lengths for the elements in S.
 In this study, we will mainly express the given above invariants of a special pseudo-symmetric numerical semigroup family in terms of its generators.

Factorization invariants of the additive structure of exponential Puiseux semirings

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

Length-factoriality in commutative monoids and integral domains

An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x∈M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, length-factoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is, in general, weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M, and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M. Then we study the connection between length-factoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely long and purely short irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.

Betti elements and catenary degree of telescopic numerical semigroup family

The catenary degree is an invariant that measures the distance between factorizations of elements within a numerical semigroup. In general, all possible catenary degrees of the elements of the numerical semigroups occur as the catenary degree of one of its Betti elements. In this study, Betti elements of some telescopic numerical semigroup families with embedding dimension three were found and formulated. Then, with the help of these formulas, Frobenius numbers and genus of these families were obtained. Also, the catenary degrees of telescopic numerical semigroups were found with the help of factorizations of Betti elements of these semigroups.

Presentations of a numerical semigroup

In this paper, we mainly study the minimal presentations of numerical semigroups. Moreover, we examine the concept of gluing, complete intersection, catenary degree, elasticity of some numerical semigroups. Â

Factorization theory in commutative monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

The monotone catenary degree of monoids of ideals

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.

Minimal relations and catenary degrees in Krull monoids

Let $H$ be a Krull monoid with class group $G$. Then, $H$ is factorial if and only if $G$ is trivial. Sets of lengths and sets of catenary degrees are well studied invariants describing the arithmetic of $H$ in the non-factorial case. In this note, we focus on the set Ca $(H)$ of catenary degrees of $H$ and on the set $\mathcal R (H)$ of distances in minimal relations. We show that every finite nonempty subset of $\mathbb{N} _{\ge 2}$ can be realized as the set of catenary degrees of a Krull monoid with finite class group. This answers F. Halter-Koch [Problem 4.1]{N-P-T-W16a}. Suppose, in addition, that every class of $G$ contains a prime divisor. Then, Ca $(H)\subset \mathcal R (H)$ and $\mathcal R (H)$ contain a long interval. Under a reasonable condition on the Davenport constant of $G$, $\mathcal R (H)$ coincides with this interval, and the maximum equals the catenary degree of $H$.

The computation of factorization invariants for affine semigroups

We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

Sets of arithmetical invariants in transfer Krull monoids

Transfer Krull monoids are a recently introduced class of monoids and include the multiplicative monoids of all commutative Krull domains as well as of 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.

Arithmetical invariants of local quaternion orders

Let $D$ be a DVR, let $K$ be its quotient field, and let $R$ be a $D$-order in a quaternion algebra $A$ over $K$. The elasticity of $R^\bullet$ is $\rho(R^\bullet) = \sup\{\, k/l : u_1\cdots u_k = v_1 \cdots v_l \text{ with $u_i$, $v_j$ atoms of $R^\bullet$ and $k$, $l \ge 1$} \,\}$ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of $\rho(R^\bullet)$ and show that the set of distances $\Delta(R^\bullet)$ and all catenary degrees $\mathsf c_\mathsf d(R^\bullet)$ are finite. In the setting of noncommutative orders in central simple algebras, such results have only been understood for hereditary orders and for a few individual examples.

REALISABLE SETS OF CATENARY DEGREES OF NUMERICAL MONOIDS

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.

The catenary degrees of elements in numerical monoids generated by arithmetic sequences

ABSTRACTWe compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.

On factorization invariants and Hilbert functions

Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical semigroups (additive subsemigroups of the natural numbers), several factorization invariants are known to admit predictable behavior for sufficiently large semigroup elements. In particular, the catenary degree and delta set invariants are both eventually periodic, and the omega-primality invariant is eventually quasilinear. In this paper, we demonstrate how each of these invariants is determined by Hilbert functions of graded modules. In doing so, we extend each of the aforementioned eventual behavior results to finitely generated semigroups, and provide a new framework through which to study factorization structures in this setting.

Numerical Semigroups on Compound Sequences

We generalize the geometric sequence {ap, ap−1b, ap−2b2,…, bp} 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.

On the set of catenary degrees of finitely generated cancellative commutative monoids

The catenary degree of an element [Formula: see text] of a cancellative commutative monoid [Formula: see text] is a nonnegative integer measuring the distance between the irreducible factorizations of [Formula: see text]. The catenary degree of the monoid [Formula: see text], defined as the supremum over all catenary degrees occurring in [Formula: see text], has been studied as an invariant of nonunique factorization. In this paper, we investigate the set [Formula: see text] of catenary degrees achieved by elements of [Formula: see text], focusing on the case where [Formula: see text] is finitely generated (where [Formula: see text] is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of [Formula: see text] that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for [Formula: see text].

Measuring primality in numerical semigroups with embedding dimension three

In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree.

Factorization theory: From commutative to noncommutative settings

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the ω-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n×n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.

When the catenary degree agrees with the tame degree in numerical semigroups of embedding dimension three

Garcia Sanchez is supported by the projects MTM2010-15595, FQM-343, FQM-5849, and FEDER funds. The contents of this article are part of Viola’s master’s thesis. Part of this work was done while she visited the Univerisidad de Granada under the European Erasmus mobility program.