The Mardešić Conjecture and free products of Boolean algebras

Martínez-Cervantes, Gonzalo; Plebanek, Grzegorz

doi:10.1090/proc/14313

The Mardešić Conjecture and free products of Boolean algebras
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by Gonzalo Martínez-Cervantes and Grzegorz Plebanek

Proc. Amer. Math. Soc. 147 (2019), 1763-1772

DOI: https://doi.org/10.1090/proc/14313

Published electronically: December 19, 2018

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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 \[ L_1\times L_2\times \dots \times L_d\to K_1\times K_2\times \cdots \times K_{d}\times K_{d+1},\] 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 Mardešić.

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 in [Order 14 (1997), pp. 21–38].

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