Modular Frobenius pseudo-varieties

Abstract

If m in {mathbb {N}} setminus {0,1} and A is a finite subset of bigcup _{k in {mathbb {N}} setminus {0,1}} {1,ldots ,m-1}^k, then we denote by C(m,A)={S∈Sm∣s1+⋯+sk-m∈Sif(s1,…,sk)∈Skand(s1modm,…,skmodm)∈A}.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathscr {C}}(m,A) =&\\{ S\\in {\\mathscr {S}}_m \\mid s_1+\\cdots +s_k-m \\in S \ ext { if } (s_1,\\ldots ,s_k)\\in S^k \ ext { and } \\\\ {}&\\qquad (s_1 \\bmod m, \\ldots , s_k \\bmod m)\\in A \\}. \\end{aligned}$$\\end{document}In this work we prove that {mathscr {C}}(m,A) is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to {mathscr {C}}(m,A) and to compute all the elements of {mathscr {C}}(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 {mathscr {C}}(m,A) when A={1,ldots ,m-1}^3, A={(1,1),ldots ,(m-1,m-1)}, and A={1,ldots ,m-1}^2 setminus {(1,1),ldots ,(m-1,m-1)}, respectively.