p-Numerical semigroups with p-symmetric properties

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