SYMMETRIES ON ALMOST SYMMETRIC NUMERICAL SEMIGROUPS HIROKATSU NARI (NIHON UNIVERSITY)

Let N be the set of nonnegative integers. A numerical semigroup H is a subset of N which is closed under addition and N \ H is a finite set. Every numerical semigroup H admits a finite system of generators, that is, there exist a1 , ..., an ∈ H such that H = ha1 , ..., an i = {λ1 a1 + · · · + λn an | λ1 , ..., λn ∈ N}. We always assume that 0 ∈ H. We define F(H) = max{n | n 6∈ H} and g(H) = Card(N \ H). We call F(H) the Frobenius number of H, and we call g(H) the genus of H. We say that an integer x is a pseudo-Frobenius number of H if x 6∈ H and x + h ∈ H for all h ∈ H, h 6= 0. We will denote by PF(H) the set of pseudo-Frobenius numbers of H, and its cardinality is the type of H, denoted by t(H). If H = ha1 , a2 , . . . , an i, then we call a1 the multiplicity of H and denote it by m(H), and we call n the embedding dimension of H and denote it by e(H). For a numerical semigroup H with maximal ideal M = H \ {0}, we set M − M = {x ∈ N | x + M ⊆ M } and K = {F(H) − z | z 6∈ H}. We call M − M the dual of M and denote it by H ∗ , and we call K the canonical ideal of H. Definition 1. [BF] We say that a numerical semigroup H is almost symmetric if K \ H = PF(H) \ {F(H)}. Theorem 2. [Ba], [BF] Let H be a numerical semigroup with maximal ideal M . Then the following conditions are equivalent. (1) H is almost symmetric. (2) K ⊂ H ∗ . (3) 2 g(H) = F(H) + t(H). In this talk, we will characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. Moreover we give a criterion for H ∗ to be almost symmetric numerical semigroup. Our first result is the following theorem. Theorem 3. Let H be a numerical semigroup and let PF(H) = {f1 < f2 < · · · < ft(H) = F(H)}. Then the following conditions are equivalent. (1) H is almost symmetric. (2) z 6∈ H implies that either F(H)−z ∈ H or z = fi for all i ∈ {1, 2, ..., t(H)−1}. (3) fi + ft(H)−i = F(H) for all i ∈ {1, 2, . . . , t(H) − 1}. The following is the key lemma to prove our second result. 1

2

HIROKATSU NARI (NIHON UNIVERSITY)

Lemma 4. Let H be a numerical semigroup. Then F(H ∗ ) = F(H) − m(H). Theorem 5. Let H be an almost symmetric numerical semigroup. Then H ∗ is almost symmetric if and only if m(H) = t(H) + t(H ∗ ). Corollary 6. Let H be an almost symmetric numerical semigroup with t(H) ≤ 2. Then H ∗ is almost symmetric if and only if e(H) = m(H) − 1. References [Ba]

[BF] [RG]

V. Barucci, On propinquity of numerical semigroups and one-dimensional local Cohen Macaulay rings, Commutative algebra and its applications, 49-60, Walter de Gruyter, Berlin, 2009. V. Barucci, R. Fr¨ oberg, One-dimensional almost Gorenstein rings, J. Algebra, 188 (1997), 418-442. J. C. Rosales, P. A. Garc´ıa-S´anchez, Numerical semigroups, Springer Developments in Mathematics, Volume 20 (2009).

SYMMETRIES ON ALMOST SYMMETRIC NUMERICAL ...

Frobenius number of H, and we call g(H) the genus of H. We say that an integer x ... will denote by PF(H) the set of pseudo-Frobenius numbers of H, and its ...

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