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The Mardešić Conjecture and free products of Boolean algebras


Authors: Gonzalo Martínez-Cervantes and Grzegorz Plebanek
Journal: Proc. Amer. Math. Soc. 147 (2019), 1763-1772
MSC (2010): Primary 54F05, 54F45; Secondary 03G05, 06E15
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

$\displaystyle L_1\times L_2\times \dots \times L_d\to K_1\times K_2\times \cdots \times K_{d}\times K_{d+1},$

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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Additional Information

Gonzalo Martínez-Cervantes
Affiliation: Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: gonzalo.martinez2@um.es

Grzegorz Plebanek
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: grzes@math.uni.wroc.pl

DOI: https://doi.org/10.1090/proc/14313
Keywords: Linearly ordered topological space, interval algebra, free product
Received by editor(s): December 5, 2017
Received by editor(s) in revised form: December 9, 2017, and July 2, 2018
Published electronically: December 19, 2018
Additional Notes: The first author was partially supported by the research project 19275/PI/14 funded by Fundación Séneca – Agencia de Ciencia y Tecnología de la Región de Murcia within the framework of PCTIRM 2011-2014 and by Ministerio de Economía y Competitividad and FEDER (project MTM2014-54182-P)
This research was done during the second’s author stay at Facultad de Matemáticas, Universidad de Murcia, supported by Fundación Séneca – Agencia de Ciencia y Tecnología de la Región de Murcia through its regional programme Jiménez de la Espada
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2018 American Mathematical Society