| Abstrakt: |
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 -pseudo-symmetric 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. [ABSTRACT FROM AUTHOR] |
