The Tree of Good Semigroups in $${\mathbb {N}}^2$$ N 2 and a Generalization of the Wilf Conjecture
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The Tree of Good Semigroups in N2 and a Generalization of the Wilf Conjecture Nicola Maugeri
and Giuseppe Zito
Abstract. Good subsemigroups of Nd have been introduced as the most natural generalization of numerical ones. Although their definition arises by taking into account the properties of value semigroups of analytically unramified rings (for instance the local rings of an algebraic curve), not all good semigroups can be obtained as value semigroups, implying that they can be studied as pure combinatorial objects. In this work, we are going to introduce the definition of length and genus for good semigroups in Nd . For d = 2, we show how to count all the local good semigroups with a fixed genus through the introduction of the tree of local good subsemigroups of N2 , generalizing the analogous concept introduced in the numerical case. Furthermore, we study the relationships between these elements and others previously defined in the case of good semigroups with two branches, as the type and the embedding dimension. Finally, we show that an analogue of Wilf’s conjecture fails for good semigroups in N2 . Mathematics Subject Classification. 13A18, 14H99, 13H99, 20M25. Keywords. Good semigroups, genus of a good semigroup, type of a good semigroup, Wilf conjecture.
1. Introduction The study of good semigroups was formerly motivated by the fact that they are the value semigroups of one-dimensional analytically unramified rings (such as the local rings of an algebraic curve). The definition appeared the Part of this work was done while the first author was visiting the Universities of Almeria and Granada supported by the project MTM2014-55367-P, which is funded by Ministerio de Econom´ıa y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andaluc´ıa Grant Number FQM-343. Both the authors were funded by the project “Propriet` a algebriche locali e globali di anelli associati a curve e ipersuperfici” PTR 2016-18—Dipartimento di Matematica e Informatica, Universit` a di Catania’’ . Nicola Maugeri and Giuseppe Zito contributed equally to this work.
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first time in [1] and these objects were widely studied in several works [2, 5,10,11,17]. In [1], the authors proved that the class of good semigroups is actually larger than the one of value semigroups. Thus, such semigroups can be seen as a natural generalization of numerical semigroups and can be studied using a more combinatorial approach without necessarily referring to the ring theory context. In recent works [8,9,19], some notable elements and properties of numerical semigroups have been generalized to the case of good semigroups. The main purpose of this work is to generalize the definitions of length and genus of an ideal of a numerical semigroup to the case of good ideals of a good semigroup studying also the relationships between them and the other objects defined in the previous works in the case of subsemigroups of N2 . If R is an analytically unramified ring, th
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