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Paper   IPM / M / 15375
School of Mathematics
  Title:   Primary decomposition of ideals of lattice homomorphisms
  Author(s):  Leila Sharifan
  Status:   Published
  Journal: Electron. J. Combin.
  Vol.:  25
  Year:  2018
  Pages:   # P3.8
  Supported by:  IPM
For two given finite lattices L and M, we introduce the ideal of lattice homomorphism J(L, M), whose minimal monomial generators correspond to lattice homomorphisms φ : L → M. We show that L is a distributive lattice if and only if the equidimensinal part of J(L, M) is the same as the equidimensional part of the ideal of poset homomorphisms I(L, M). Next, we study the minimal primary decomposition of J(L, M) when L is a distributive lattice and M = [2]. We present some methods to check if a monomial prime ideal belongs to ass(J(L, [2])), and we give an upper bound in terms of combinatorial properties of L for the height of the minimal primes. We also show that if each minimal prime ideal of J(L, [2]) has height at most three, then L is a planar lattice and width(L) 6 2. Finally, we compute the minimal primary decomposition when L = [m] × [n] and M = [2].

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