## The Mardešić Conjecture and free products of Boolean algebras

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- by Gonzalo Martínez-Cervantes and Grzegorz Plebanek PDF
- Proc. Amer. Math. Soc.
**147**(2019), 1763-1772 Request permission

## 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].

## References

- Antonio Avilés,
*Compact spaces that do not map onto finite products*, Fund. Math.**202**(2009), no. 1, 81–96. MR**2457486**, DOI 10.4064/fm202-1-4 - Antonio Avilés, Gonzalo Martínez-Cervantes, and Grzegorz Plebanek,
*Weakly Radon-Nikodým Boolean algebras and independent sequences*, Fund. Math.**241**(2018), no. 1, 45–66. MR**3756099**, DOI 10.4064/fm404-5-2017 - Lynne Baur and Lutz Heindorf,
*Initial chain algebras on pseudotrees*, Order**14**(1997), no. 1, 21–38. MR**1468953**, DOI 10.1023/A:1005830829350 - Lynne Baur,
*Cardinal functions on initial chain algebras on pseudotrees*, Order**17**(2000), no. 1, 1–21. MR**1776931**, DOI 10.1023/A:1006415818082 - W. Bula, W. Dębski, and W. Kulpa,
*A short proof that a compact ordered space cannot be mapped onto a nonmetric product*, Proc. Amer. Math. Soc.**82**(1981), no. 2, 312–313. MR**609675**, DOI 10.1090/S0002-9939-1981-0609675-3 - Lutz Heindorf,
*On subalgebras of Boolean interval algebras*, Proc. Amer. Math. Soc.**125**(1997), no. 8, 2265–2274. MR**1389523**, DOI 10.1090/S0002-9939-97-03851-3 - Sibe Mardešić,
*Mapping products of ordered compacta onto products of more factors*, Glasnik Mat. Ser. III**5(25)**(1970), 163–170 (English, with Serbo-Croatian summary). MR**275378** - Sibe Mardešić,
*Continuous images of linearly ordered continua and compacta*, Topology Appl.**195**(2015), 34–49. MR**3414873**, DOI 10.1016/j.topol.2015.09.029 - L. B. Treybig,
*Concerning continuous images of compact ordered spaces*, Proc. Amer. Math. Soc.**15**(1964), 866–871. MR**167953**, DOI 10.1090/S0002-9939-1964-0167953-9 - A. J. Ward,
*Some properties of images of ordered compacta, with special reference to topological limits*, unpublished (1965).

## 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
- MR Author ID: 239421
- Email: grzes@math.uni.wroc.pl
- 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
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3910440