The notion of almost symmetric numerical semigroup was given by V. Barucci and R. Fröberg in [BF]. We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for H * (the dual of M ) to be almost symmetric numerical semigroup. Using these results we give a formula for multiplicity of an opened modular numerical semigroups. Finally, we show that if H 1 or H 2 is not symmetric, then the gluing of H 1 and H 2 is not almost symmetric.

Let H = a, b, c be a numerical semigroup generated by three elements and let R = k[H] be its semigroup ring over a field k. We assume H is not symmetric and assume that the definig ideal of R is defined by maximal minorsThen we will show that the genus of H is determined by the Frobenius number F(H) and αβγ or α ′ β ′ γ ′ . In particular, we show that H is pseudo-symmetric if and only if αβγ = 1 or α ′ β ′ γ ′ = 1.Also, we will give a simple algorithm to get all the pseudo-symmetric numerical semigroups H = a, b, c with give Frobenius number.

No abstract

No abstract