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The covariety of perfect numerical semigroups with fixed Frobenius number

María Ángeles Moreno-Frías, José Carlos Rosales (2024)

Czechoslovak Mathematical Journal

Let S be a numerical semigroup. We say that h ∈ ℕ ∖ S is an isolated gap of S if { h - 1 , h + 1 } ⊆ S . A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by m ( S ) the multiplicity of a numerical semigroup S . A covariety is a nonempty family 𝒞 of numerical semigroups that fulfills the following conditions: there exists the minimum of 𝒞 , the intersection of two elements of 𝒞 is again an element of 𝒞 , and S ∖ { m ( S ) } ∈ 𝒞 for all S ∈ 𝒞 such that S ≠ min ( 𝒞 ) . We prove that the set 𝒫 ( F ) = { S : S is a perfect numerical semigroup with...

The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated

Maria Angeles Moreno-Frías, José Carlos Rosales (2024)

Mathematica Bohemica

Let Δ be a numerical semigroup. In this work we show that 𝒥 ( Δ ) = { I ∪ { 0 } : I is an ideal of Δ } is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set 𝒥 a ( Δ ) = { S ∈ 𝒥 ( Δ ) : max ( Δ ∖ S ) = a } for a given a ∈ Δ . As a consequence, we obtain another algorithm that computes all the elements of 𝒥 ( Δ ) with a fixed genus.

The rank of a commutative semigroup

Antonio M. Cegarra, Mario Petrich (2009)

Mathematica Bohemica

The concept of rank of a commutative cancellative semigroup is extended to all commutative semigroups S by defining rank S as the supremum of cardinalities of finite independent subsets of S . Representing such a semigroup S as a semilattice Y of (archimedean) components S α , we prove that rank S is the supremum of ranks of various S α . Representing a commutative separative semigroup S as a semilattice of its (cancellative) archimedean components, the main result of the paper provides several characterizations...

The tame degree and related invariants of non-unique factorizations

Franz Halter-Koch (2008)

Acta Mathematica Universitatis Ostraviensis

Local tameness and the finiteness of the catenary degree are two crucial finiteness conditions in the theory of non-unique factorizations in monoids and integral domains. In this note, we refine the notion of local tameness and relate the resulting invariants with the usual tame degree and the ω -invariant. Finally we present a simple monoid which fails to be locally tame and yet has nice factorization properties.

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