Numerical semigroups in a problem about cost-effective transport

Abstract

Abstract Let ℕ ${{\mathbb{N}}}$ be the set of nonnegative integers. A problem about how to transport profitably an organized group of persons leads us to study the set T formed by the integers n such that the system of inequalities, with nonnegative integer coefficients, a 1 ⁢ x 1 + ⋯ + a p ⁢ x p < n < b 1 ⁢ x 1 + ⋯ + b p ⁢ x p $a_{1}x_{1}+\cdots+a_{p}x_{p}<n<b_{1}x_{1}+\cdots+b_{p}x_{p}$ has at least one solution in ℕ p ${{\mathbb{N}}^{p}}$ . We will see that T ∪ { 0 } ${T\cup\{0\}}$ is a numerical semigroup. Moreover, we will show that a numerical semigroup S can be obtained in this way if and only if { a + b - 1 , a + b + 1 } ⊆ S ${\{a+b-1,a+b+1\}\subseteq S}$ , for all a , b ∈ S ∖ { 0 } ${a,b\in S\setminus\{0\}}$ . In addition, we will demonstrate that such numerical semigroups form a Frobenius variety and we will study this variety. Finally, we show an algorithmic process in order to compute T.