Embedding coproducts of partition lattices

TL;DR

This paper demonstrates that the lattice of all equivalence relations on an infinite set contains complex coproduct structures, including a coproduct of many copies of itself, answering a longstanding question in lattice theory.

Contribution

It proves that Eq(X) contains a 0,1-sublattice isomorphic to the coproduct of two copies of itself, extending to many copies, thus advancing understanding of its algebraic structure.

Findings

Eq(X) contains the 0,1-sublattice as a coproduct of two copies.

Eq(X) contains the coproduct of 2^{card(X)} copies of itself.

The results answer a question posed by G.M. Bergman.

Abstract

We prove that the lattice Eq(X) of all equivalence relations on an infinite set X contains, as a 0,1-sublattice, the 0-coproduct of two copies of itself, thus answering a question by G.M. Bergman. Hence, by using methods initiated by de Bruijn and further developed by Bergman, we obtain that Eq(X) also contains, as a sublattice, the coproduct of 2^{card(X)} copies of itself.