$p$-numerical semigroups with $p$-symmetric properties

Abstract

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation a 1 x 1 + • • • + a k x k = n (a 1 , . . . , a k are given positive integers with gcd(a 1 , . . . , a k ) = 1) does not have a non-negative integer solution (x 1 , . . . , x k ). The generalized Frobenius number (called the p-Frobenius number) is the largest integer such that this linear equation has at most p solutions. That is, when p = 0, the 0-Frobenius number is the original Frobenius number.In this paper, we introduce and discuss p-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer p, p-gaps, p-symmetric semigroups, p-pseudosymmetric semigroups, and the like are defined, and their properties are obtained. When p = 0, they correspond to the original gaps, symmetric semigroups, and pseudo-symmetric semigroups, respectively.