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On the Coextension of Cut-Continuous Pomonoids
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  • Published: 02 August 2018

On the Coextension of Cut-Continuous Pomonoids

  • David Kruml1,
  • Jan Paseka1 &
  • Thomas Vetterlein2 

Order volume 36, pages 271–290 (2019)Cite this article

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Abstract

We introduce cut-continuous pomonoids, which generalise residuated posets. The latter’s defining condition is that the monoidal product is residuated in each argument; we define cut-continuous pomonoids by requiring that the monoidal product is in each argument just cut-continuous. In the case of a total order, the condition of cut-continuity means that multiplication distributes over existing suprema. Morphisms between cut-continuous pomonoids can be chosen either in analogy with unital quantales or with residuated lattices. Under the assumption of commutativity and integrality, congruences are in the latter case induced by filters, in the same way as known for residuated lattices. We are interested in the construction of coextensions: given cut-continuous pomonoids K and C, we raise the question how we can determine the cut-continuous pomonoids L such that C is a filter of L and the quotient of L induced by C is isomorphic to K. In this context, we are in particular concerned with tensor products of modules over cut-continuous pomonoids. Using results of M. Erné and J. Picado on closure spaces, we show that such tensor products exist. An application is the construction of residuated structures related to fuzzy logics, in particular left-continuous t-norms.

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Acknowledgments

Open access funding provided by Austrian Science Fund (FWF). The authors acknowledge the support by the Czech Science Foundation (GAČR): project 15-34697L and the Austrian Science Fund (FWF): project I 1923-N25.

They would moreover like to express their gratitude to the anonymous reviewers whose thorough and detailed comments lead to a considerable improvement of this paper. Finally, the authors are indepted to M. Erné whose article [8] helped to enhance the results of this paper in a substantial manner.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    David Kruml & Jan Paseka

  2. Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Altenberger Straße 69, 4040, Linz, Austria

    Thomas Vetterlein

Authors
  1. David Kruml
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  2. Jan Paseka
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  3. Thomas Vetterlein
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Corresponding author

Correspondence to Thomas Vetterlein.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Kruml, D., Paseka, J. & Vetterlein, T. On the Coextension of Cut-Continuous Pomonoids. Order 36, 271–290 (2019). https://doi.org/10.1007/s11083-018-9466-3

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  • Received: 09 June 2017

  • Accepted: 11 July 2018

  • Published: 02 August 2018

  • Issue Date: 15 July 2019

  • DOI: https://doi.org/10.1007/s11083-018-9466-3

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Keywords

  • Partially ordered monoid
  • Cut-continuous pomonoid
  • Residuated poset
  • Coextension of cut-continuous pomonoids
  • Tensor product of modules over cut-continuous pomonoids
  • Closure space
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