Numerical Semigroups on Compound Sequences

Abstract

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.

On Sylvester sums of compound sequence semigroup complements

“…Remark 2.5. We note that our definition of compound sequence differs slightly from the definition given in [3], where there is the additional condition that 2 ≤ a i < b i for all i. The following proposition is proved in their paper, though this additional condition isn't used in the proof so the result holds for our definition as well.…”

“…Compound sequences are generalizations of geometric sequences, which occur when a 1 = · · · = a k and b 1 = · · · = b k . Numerical semigroups arising from compound sequences have been studied in [3], following work on numerical semigroups from geometric sequences in [6] and [19].…”

Characterizations of numerical semigroup complements via Apéry sets

“…For compound sequences, we follow the notation of [4]. See also [5]. Compound sequences are special cases of smooth sequences (by Corollary 3.11), so we apply the above results to compound sequences and obtain an alternate proof of [4, Theorem 3.3].…”

Elasticity in Apéry Sets

“…A numerical semigroup S is a subset of Z ≥0 with finite complement that is closed under + and contains 0. Numerical semigroups (which are, in fact, monoids, since they contain 0) have been the subject of considerable recent study [9,12,[15][16][17][18]. Many applications are known, such as in coding theory [7], algebraic geometry [6], and discrete optimization [19].…”