An equational basis for the variety generated by the class of partition lattices

Question

I define a partition lattice to be a bounded lattice $(L;\land,\vee,0,1)$ that is isomorphic to the lattice of all partitions of some set $S$. What is an equational basis for the variety generated by the class of partition lattices? Is it simply the variety of bounded lattices, or is it some restricted subvariety of the variety of all bounded lattices? In other words, I am really asking if partition lattices satisfy additional equations beyond that of bounded lattices.

Answer 1

I am really asking if partition lattices satisfy additional equations beyond that of bounded lattices.

No. To see this, it suffices to show that any identity in the language of bounded lattices that holds in every partition lattice must hold in every bounded lattice.

Assume that $\varepsilon$ is an identity that holds in every partition lattice. Let $L$ be any bounded lattice. It follows from the main theorem of

P. Whitman
Lattices, equivalence relations and subgroups
Bull. Amer. Math. Soc. 52 (1946), 507-522.

that every (not-necessarily-bounded) lattice is isomorphic to a (not-necessarily-bounded) sublattice of a partition lattice. With very little work, one can establish a result for bounded lattices from this: Every bounded lattice is a bound-preserving homomorphic image of a bounded sublattice of a partition lattice. Hence $L$ is a homomorphic image of a sublattice of some partition lattice $\Pi$. Identities are preserved under the formation homomorphic images and sublattices, so $L$ must satisfy $\varepsilon$. This explains why any identity in the language of bounded lattices that holds in all partition lattices must hold in any bounded lattice.

Here is the "very little work" I mentioned above. Let $L$ be any bounded lattice with bottom element $0^L$ and top element $1^L$. Let $f\colon L\to \Pi$ be the Whitman embedding of $L$ into a partition lattice $\Pi$. Let $0^{\Pi}$ and $1^{\Pi}$ be the bottom and top of $\Pi$. Let $K$ be the bounded sublattice of $\Pi$ whose universe is $f(L)\cup \{0^{\Pi}, 1^{\Pi}\}$. There is a surjective, bound-preserving homomorphism $g\colon K\to L$ defined by $g(0^{\Pi})=0^L, g(1^{\Pi})=1^L$ and $g(x)=f^{-1}(x)$ for every $x\in f(L)$. This represents $L$ has a bound-preserving homomorphic image of a bounded sublattice of $\Pi$.