Modular Frobenius pseudo-varieties

Abstract

If m is an element of N \ (0, 1) and A is a finite subset of boolean OR(k is an element of N\{0,1}) {1, ..., m - 1}(k), then we denote by

l(m, A) ={S is an element of S-m vertical bar s(1) + ... + s(k) - m is an element of S if (s(1), ..., s(k)) is an element of S-k and

(s(1 )mod m, ..., s(k) mod m) is an element of A}.

In this work we prove that l(m, A) is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to l(m, A) and to compute all the elements of l(m, A) with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to l(m, A) when A = {1, ..., m - 1}(3), A = {(1, 1), ..., (m - 1, m - 1)}, and A = {1, ..., m - 1)(2)\{(1, 1), ..., (m - 1, m - 1)}, respectively.