Free Boolean algebras over unions of two well orderings

TL;DR

This paper investigates free Boolean algebras generated by unions of two well orderings, classifies certain topological sublattices, and demonstrates the existence of many non-isomorphic nearly ordinal algebras.

Contribution

It introduces nearly ordinal algebras over unions of well orderings, classifies sublattices of specific products, and answers a question about the diversity of such algebras.

Findings

Existence of 2^κ non-isomorphic nearly ordinal algebras for each uncountable κ.

Only ℵ_1 many closed sublattices of (ω_1+1)×(ω_1+1).

2^ℵ_1 many topological types of closed subsets of the Tikhonov plank.

Abstract

Given a partially ordered set $P$ there exists the most general Boolean algebra $F(P)$ which contains $P$ as a generating set, called the {\it free Boolean algebra} over $P$. We study free Boolean algebras over posets of the form $P=P_0\cup P_1$, where $P_0,P_1$ are well orderings. We call them {\it nearly ordinal algebras}. Answering a question of Maurice Pouzet, we show that for every uncountable cardinal $\kappa$ there are $2^\kappa$ pairwise non-isomorphic nearly ordinal algebras of cardinality $\kappa$. Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product $(\omega_1+1)\times(\omega_1+1)$, thus showing that there are only $\aleph_1$ many of them. In contrast with the last result, we show that there are $2^{\aleph_1}$ topological types of closed subsets of the Tikhonov plank $(\omega_1+1)\times(\omega+1)$.