A categorical view of varieties of ordered algebras. (10th April 2022)

• Record Type: Journal Article
• Authors: Adámek, J.; Dostál, M.; Velebil, J.
• Is Part Of: Mathematical structures in computer science. Volume 32:Number 4(2022)
• Page Start: 349
• Page End: 373
• Publication Date: 2022-04-10
• Subjects: Ordered algebras -- varieties -- strongly finitary monads; Computer science -- Mathematics -- Periodicals; 004.015105
• DOI: 10.1017/S0960129521000463
• Languages: English
• ISSNs: 0960-1295

Abstract

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 $\mathsf{Pos}$ .