Infinite families of 3-numerical semigroups with arithmetic-like links

Abstract

Let S = a, b,N be a numerical semigroup generated by a, b,N ∈ N with 1 < a < b < N and gcd(a, b,N) = 1. The conductor of S, denoted by c(S) or c(a, b,N), is the minimum element of S such that c(S) + m ∈ S for all m ∈ N ∪ {0}. Some arithmetic-like links between 3-numerical semigroups were remarked by V. Arnold. For instance he gave links of the form $\displaystyle\frac{c(13, 32, 52)}{c(13, 33, 51)} = \displaystyle\frac{c(9, 43, 45) }{c(9, 42, 46)} = \displaystyle\frac{c(5, 35, 37)}{c(5, 34, 38)} = 2 or \displaystyle\frac{c(4, 20, 73)}{c(4, 19, 74)} = 4$. In this work several infinite families of 3-numerical semigroups with similar properties are given. These families have been found using a plane geometrical approach, known as L-shaped tile, that can be related to a 3-numerical semigroup. This tile defines a plane tessellation that gives information on the related semigroup.