Marde\v{s}i\'c conjecture and free products of Boolean algebras

TL;DR

This paper proves a conjecture related to the structure of products of linearly ordered compact spaces and Boolean algebras, showing conditions under which certain factors are metrizable and addressing open problems in the field.

Contribution

It establishes a new result confirming Mardešić's conjecture and provides insights into Boolean algebras without large free products, solving related open problems.

Findings

If a product of linearly ordered compact spaces maps onto a product with an additional factor, some factors are metrizable.

The paper answers a problem posed by Mardešić regarding product spaces.

It presents results on Boolean algebras without large free products, addressing Baur and Heindorf's problem.

Abstract

We show that for every $d\ge 1$, if $L_1,\ldots, L_d$ are linearly ordered compact spaces and there is a continuous surjection \[ L_1\times L_2\times \dots\times L_d\to K_1\times K_2\times\ldots\times K_{d}\times K_{d+1},\] where all the spaces $K_i$ are infinite, then $K_i, K_j$ are metrizable for some $1\le i<j\le d+1$. This answers a problem posed by S. Marde\v{s}i\'c. We present some related results on Boolean algebras not containing free products with too many uncountable factors. In particular, we answer a problem on initial chain algebras that was posed by L. Baur and L. Heindorf.