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Algebra Seminar talk

2009-01-23
Jonathan Farley
Functions on Distributive Lattices with the Congruence Substitution Property: Some Problems of Grätzer from 1964

Abstract:
Let L be a bounded distributive lattice and let k≥1. A function f:L^k→L has the congruence substitution property if, for every congruence θ of L, and all (a₁,b₁),...,(a_k,b_k)∈θ, we have f(a₁,...,a_k) θ f(b₁,....,b_k). The set of all such functions forms a bounded distributive lattice, denoted S_k(L) (also called the lattice of Boolean functions). Let S(L) be the lattice of all Boolean functions of finite arity on the variables x₁, x₂, ....

In 1964, Grätzer asked:

• Question 1. Let L and M be bounded

  distributive lattices such that S₁(L)≅S₁(M). Is S_k(L) necessarily isomorphic to S_k(M)?

• Question 2. Characterize those lattices

  isomorphic to S_k(L) or S(L) for some bounded distributive lattice L.

Using Priestley duality, we answer both questions. (The corresponding questions in the unbounded case---also asked by Grätzer---are open.)

You have to come to the talk to see what the answer is.