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A categorical view of varieties of ordered algebras

Published online by Cambridge University Press:  10 January 2022

J. Adámek ,
M. Dostál Open the ORCID record for M. Dostál [Opens in a new window]  and
J. Velebil
J. Adámek
Affiliation:
Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic Institute for Theoretical Computer Science, Technical University Braunschweig, Germany
M. Dostál*
Affiliation:
Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic
J. Velebil
Affiliation:
Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic
*
*Corresponding author. Email: [email protected]
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Abstract

It is well known that classical varieties of $\Sigma$ -algebras correspond bijectively to finitary monads on $\mathsf{Set}$ . We present an analogous result for varieties of ordered $\Sigma$ -algebras, that is, categories of algebras presented by inequations between $\Sigma$ -terms. We prove that they correspond bijectively to strongly finitary monads on $\mathsf{Pos}$ . That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power. We also show that these monads are liftings of finitary monads on $\mathsf{Set}$ . Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on $\mathsf{Set}$ to strongly finitary monads on $\mathsf{Pos}$ .

Keywords

Ordered algebrasvarietiesstrongly finitary monads
Type
Special Issue: The Power Festschrift
Information
Mathematical Structures in Computer Science , Volume 32 , Special Issue 4: The Power Festschrift , April 2022 , pp. 349 - 373
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

1

J. Adámek and M. Dostál acknowledge the support of the grant No. 19-0092S of the Czech Grant Agency.

References

Adámek, J., Ford, C., Milius, S. and Schröder, L. (2021). Finitary monads on the category of posets. arXiv:2011.14796, to appear in Math. Struct. Comput. Sci. Google Scholar
Adámek, J., Herrlich, H. and Strecker, G. E. (1990). Abstract and Concrete Categories, John Wiley and Sons.Google Scholar
Adámek, J. and Rosický, J. (1994). Locally Presentable and Accessible Categories, Cambridge University Press.CrossRefGoogle Scholar
Adámek, J., Rosicky, J. and Vitale, E. (2010). What are sifted colimits? Theory and Applications of Categories 23 251–260.Google Scholar
Barr, M. (1970). Coequalizers and free triples, Mathematische Zeitschrift 116 307–322.CrossRefGoogle Scholar
Barr, M. and Wells, Ch. (1985). Toposes, Triples and Theories, Springer.CrossRefGoogle Scholar
Bloom, S. L. (1976). Varieties of ordered algebras, Journal of Computer and System Sciences 13 200–212.CrossRefGoogle Scholar
Bourke, J. (2010). Codescent objects in 2-dimensional universal algebra, PhD Thesis, University of Sydney.Google Scholar
Bourke, J. and Garner, R. (2019) Monads and theories, Advances in Mathematics 351 1024–1071.CrossRefGoogle Scholar
Blyth, T. S. (2005). Lattices and Ordered Algebraic Structures, Springer.Google Scholar
Dubuc, E. (1970). Kan Extensions in Enriched Category Theory, Springer.CrossRefGoogle Scholar
Ford, C., Milius, S. and Schröder, L. (2021). Monads on categories of relational structures, arXiv:2107.03880.Google Scholar
Golan, J. S. (2003). Partially-ordered semirings. In: Semirings and Affine Equations Over Them, Springer, 27–38Google Scholar
Kelly, G. M. (1982). Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Notes Series, vol. 64, Cambridge Univ. Press, 1982, also available as Reprints in Theory and Applications of Categories 10 (2005)Google Scholar
Kelly, G. M. (1982). Structures defined by finite limits in the enriched context I, Cahiers de Topologie et Géométrie Différentielle Catégoriques XXIII(1) 3–42.Google Scholar
Kelly, G. M. and Lack, S. (1993). Finite-product-preserving functors, Kan extensions and strongly-finitary 2-monads, Applied Categorical Structures 1 85–94.CrossRefGoogle Scholar
Kelly, G. M. and Power, A. J. (1993). Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, Journal of Pure and Applied Algebra 89 163–179 CrossRefGoogle Scholar
Kurz, A. and Rosický, J. (2017). Strongly complete logics for coalgebras, Logical Methods in Computer Science 8(3) 1–32 Google Scholar
Kurz, A. and Velebil, J. (2017). Quasivarieties and varieties of ordered algebras: Regularity and exactness, Logical Methods in Computer Science 27 1153–1194.Google Scholar
Lack, S. and Rosický, J. (2011). Notions of Lawvere theories, Applied Categorical Structures 19(1) 363–391.CrossRefGoogle Scholar
Lack, S. (1999). On the monadicity of finitary monads, Journal of Pure and Applied Algebra 140 66–73.CrossRefGoogle Scholar
Lawvere, F. W. (2004). Functorial semantics of algebraic theories, PhD Thesis, Columbia University 1963, available as Reprints in Theory and Applications of Categories 5 1–121.CrossRefGoogle Scholar
Mac Lane, S. (1988). Categories for the Working Mathematician, 2nd ed., Springer.Google Scholar
Power, J. (2005). Discrete Lawvere Theories , Lecture Notes Computer Science, vol. 3629, Springer, 348–363.Google Scholar
Rosický, J. (2012). Metric Monads, arXiv:2012.14641.Google Scholar
Trnková, V., Adámek, J., Koubek, V. and Reiterman, J. (1975). Free algebras, input processes and free monads, Commentationes Mathematicae Universitatis Carolinae 16 339–351.Google Scholar