Join-Meet Ideals of Finite Lattices

Abstract

One of the most natural classes of binomial ideals arising from combinatorics is the class of join-meet ideals of finite lattices. The purpose of the present chapter is mainly to study Grobner bases of join-meet ideals. In Section 6.1, we collect fundamental definitions and basic results on classical lattice theory. Especially, a complete proof of the characterization of distributive lattices due to Dedekind is supplied. The algebraic theory of join-meet ideals, which originated in the study on those ideals of finite distributive lattices, is introduced in Section 6.2. The highlight is the fact that the join-meet ideal of a finite lattice is a prime ideal if and only if the lattice is distributive. Furthermore, with respect to a certain reverse lexicographic order, it is shown that the set of binomial generators of the join-meet ideal of a finite lattice is a Grobner basis of the ideal if and only if the lattice is distributive. We then devote Section 6.3 to the discussion of join-meet ideals of finite non-distributive modular lattices. Furthermore, in Section 6.4, join-meet ideals of planar distributive lattices will be studied. Finally, in Section 6.5, via the theory of canonical modules and the a-invariant, projective dimension together with regularity of join-meet ideals will be discussed.