Support posets of some monomial ideals
- PDF / 1,870,461 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 204 Views
Support posets of some monomial ideals Patricia Pascual‑Ortigosa1 · E. Sáenz‑de‑Cabezón1 Received: 8 June 2020 / Accepted: 9 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The support poset of a monomial ideal I ⊆ 𝐤[x1 , … , xn ] encodes the relation between the variables x1 , … , xn and the minimal monomial generators of I. It is known that not every poset is realizable as the support poset of some monomial ideal. We describe some posets P for which we can explicitly find at least one monomial ideal IP such that P is the support poset of IP . Also, for some families of monomial ideals we describe their support posets and study their properties. As an example of application we examine the relation between forests and series-parallel ideals. Keywords Monomial ideals · Support posets · Series-parallel systems Mathematics Subject Classification 13F55
1 Introduction The support poset of a monomial ideal I ⊆ 𝐤[x1 , … , xn ] encodes the relation between the variables x1 , … , xn and the minimal monomial generators of I. It was introduced in [12] to study the set of all depolarizations of a given squarefree monomial ideal. Since many relevant features of a given monomial ideal are shared by the ideals in the same polarity class, the study of polarization and depolarization has become relevant in the last years cf. [3–5, 9, 10]. In this context, the use of the support poset is a useful tool. As it is shown in [12] not every poset is realizable as the support poset of a monomial ideal. A natural problem is therefore to find posets that can be realized as support posets of monomial ideals and provide explicit descriptions of those ideals, so that we can describe properties of the ideal based on properties of the support poset and viceversa. We address this issue in Sect. 3 of the paper in * Patricia Pascual‑Ortigosa [email protected] E. Sáenz‑de‑Cabezón eduardo.saenz‑de‑[email protected] 1
Departamento de Matemáticas y Computación, Universidad de La Rioja, La Rioja, Spain
13
Vol.:(0123456789)
P. Pascual‑Ortigosa, E. Sáenz‑de‑Cabezón
which we give some families of posets for which we can always find at least one monomial ideal supported by them (collections of lines or diamonds, and forests) and provide a full explicit description of the main features of these ideals such as their Betti numbers and free resolutions, see Propositions 3.3 and 3.6 and in particular Theorem 3.11. Another natural question related to support posets is to find a natural way to describe the support poset of some families of monomial ideals. We describe in Sect. 4 the support poset of k-out-of-n and series-parallel ideals, which correspond to relevant systems in reliability theory [11, 16, 17]. We find a particular relation between forests and series-parallel ideals, see Theorem 4.5 and Proposition 4.7. It is known that a given poset can be the support poset of several different monomial ideals. We see that this holds even within the classes of forests and series-parallel ideals, i.e. a given fo
Data Loading...
