# 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 (a_{1},b_{1}),...,(a_{k},b_{k})∈θ, we have
f(a_{1},...,a_{k}) θ f(b_{1},....,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 x_{1}, x_{2}, ....

In 1964, Grätzer asked:

*Question 1.*Let L and M be boundeddistributive lattices such that S

_{1}(L)≅S_{1}(M). Is S_{k}(L) necessarily isomorphic to S_{k}(M)?*Question 2.*Characterize those latticesisomorphic 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.