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The Free Distributive Semilattice Extension of a Distributive Poset

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Abstract

The main aim of this paper is to obtain a free distributive semilattice extension for some ordered sets satisfying a distributivity condition. That is, for an ordered set P satisfying a distributivity condition, we prove the existence of a distributive semilattice M and a monomorphism e : P ↪ M such that for every distributive semilattice L and every monomorphism f : P ↪ L there exists a unique semilattice embedding \(\widehat {f}\colon M\hookrightarrow L\) such that \(f=\widehat {f}\circ e\). To attain this, we will need to consider and study some concepts on ordered sets like filters and a distributivity condition. We consider three notions of filters on posets known in the literature, and we show some new relationships between them. We also introduce and investigate three definitions of morphisms between posets.

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Authors and Affiliations

  1. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa, Santa Rosa, Argentina

    Luciano J. González

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  1. Luciano J. González
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Correspondence to Luciano J. González.

Additional information

This work was partially supported by Universidad Nacional de La Pampa (Facultad de Ciencias Exactas y Naturales) under the grant P.I. 64 M, Res. 432/14 CD; and also by Consejo Nacional de Investigaciones Científicas y Técnicas (Argentina) under the grant PIP 112-20150-100412CO

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González, L.J. The Free Distributive Semilattice Extension of a Distributive Poset. Order 36, 359–375 (2019). https://doi.org/10.1007/s11083-018-9471-6

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  • DOI: https://doi.org/10.1007/s11083-018-9471-6

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