Algebra Seminar talk
Functions on Distributive Lattices with the Congruence Substitution Property: Some Problems of Grätzer from 1964
Let L be a bounded distributive lattice and let k≥1. A function f:Lk→L has the congruence substitution property if, for every congruence θ of L, and all (a1,b1),...,(ak,bk)∈θ, we have f(a1,...,ak) θ f(b1,....,bk). The set of all such functions forms a bounded distributive lattice, denoted Sk(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 Sk(L) necessarily isomorphic to Sk(M)?
- Question 2. Characterize those lattices
isomorphic to Sk(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.