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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: dostamat@fel.cvut.cz
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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.

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