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On Monoids in the Category of Sets and Relations

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Abstract

The category R e l is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, R e l is a monoidal category. Moreover, R e l is a locally posetal 2-category, since every homset R e l(A,B) is a poset with respect to inclusion. We examine the 2-category of monoids R e l M o n in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.

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Notes

  1. A reader who knows what the Riesz decomposition property means might wish to look at Example 12 now.

  2. The notation (C1), (C2) and (C5) is inherited from the original paper [7]. It coincides with the notation used later in several other papers and in the book [9].

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Acknowledgments

We are indebted to both anonymous referees for valuable comments and suggestions.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovak Republic

    Anna Jenčová

  2. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovak Republic

    Gejza Jenča

Authors
  1. Anna Jenčová
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  2. Gejza Jenča
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Correspondence to Gejza Jenča.

Additional information

This research is supported by grants VEGA 2/0069/16, 1/0420/15, Slovakia and by the Slovak Research and Development Agency under the contract APVV-14-0013.

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Jenčová, A., Jenča, G. On Monoids in the Category of Sets and Relations. Int J Theor Phys 56, 3757–3769 (2017). https://doi.org/10.1007/s10773-017-3304-z

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