Ideal of Lattice homomorphisms corresponding to the products of two arbitrary lattices and the lattice [2]

Abstract. Let L and M be two finite lattices. The ideal J(L,M) is a monomial ideal in a specific polynomial ring and whose minimal monomial generators correspond to lattice homomorphisms ϕ: L→M. This ideal is called the ideal of lattice homomorphism. In this paper, we study J(L,M) in the case that L is the product of two lattices L_1 and L_2 and M is the chain [2]. We first characterize the set of all lattice homomorphisms ϕ:L→[2] according to the set of all lattice homomorphisms ϕ_1:L_1→[2] and the set of all lattice homomorphisms ϕ_2:L_2→[2]. Then, by using it and the set of associated prime ideals of both J(L_1,[2] ) and J(L_2,[2]), we study the associated prime ideals of J(L,[2]). Next, we assume that L_1=[2] and we characterized ass (J (L,[2])). Then by mapping cone technique and minimal free resolution of J(L_2,[2]), we find a free resolution of J(L,[2]) and an upper bound for the projective dimension of J(L,[2]). Finally, under the above assumptions and for the case that L_2=[n], we compute the minimal free resolution of J(L,[2]).