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Quasivarieties and varieties of ordered algebras: regularity and exactness


We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets. We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense).

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The author (J. Velebil) acknowledges the support of the grant No. P202/11/1632 of the Czech Science Foundation.

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Adámek, J., Koubek, V. and Velebil, J. (2000). A duality between infinitary varieties and algebraic theories. Commentationes Mathematicae Universitatis Carolinae 41 (3) 529541.
Adámek, J. and Rosický, J. (1994). Locally Presentable and Accessible Categories, Cambridge University Press.
Adámek, J. and Rosický, J. (2001). On sifted colimits and generalized varieties. Theory Applications of Categories 8 3353.
Adámek, J., Rosický, J. and Vitale, E. (2011). Algebraic Theories, Cambridge Tracts in Mathematics volume 184, Cambridge University Press, Cambridge.
Albert, M. H. and Kelly, G. M. (1988). The closure of a class of colimits. Journal of Pure and Applied Algebra 51 117.
Barr, M., Grillet, P. A. and van Osdol, D. H. (1971). Exact Categories and Categories of Sheaves, LNM volume 236, Springer.
Bird, G. J. (1984). Limits in 2-Categories of Locally-Presented Categories, Ph.D. thesis, The University of Sydney.
Bloom, S. L. (1976). Varieties of ordered algebras. Journal of Computer and System Sciences 13 (2) 200212.
Bloom, S. L. and Wright, J. B. (1983). P-varieties — A signature independent characterization of varieties of ordered algebras. Journal of Pure and Applied Algebra 29 1358.
Bourke, J. (August 2010). Codescent Objects in 2-Dimensional Universal Algebra, Ph.D. thesis, University of Sydney.
Bourke, J. and Garner, R. (2014). Two-dimensional regularity and exactness. Journal of Pure and Applied Algebra 218 (7) 13461371.
Cohn, P. (1981). Universal Algebra, Springer.
Duskin, J. (1969). Variations on Beck's Tripleability Criterion, LNM volume 106, Springer-Verlag 74129.
El Bashir, R. and Velebil, J. (2002). Simultaneously reflective and coreflective subcategories of presheaves. Theory Applications of Categories 10 410423.
Gabriel, P. and Ulmer, F. (1971). Lokal präsentierbare Kategorien, Lecture Notes in Mathematics volume 221, Springer.
Goguen, J., Thatcher, J., Wagner, E. and Wright, J. (1977). Initial algebra semantics and continuous algebras. Journal of the ACM 24 (1) 6895.
Hyland, M. and Power, J. (2006). Discrete Lawvere theories and computational effects. Theoretical Computer Science 366 144162.
Isbell, J. (1964). Subobjects, adequacy, completeness and categories of algebras. Rozprawy Math. XXXVI 133.
Kelly, G. M. (1982). Structures defined by finite limits in the enriched context I. Cahiers de Géometrie Differentielle XXIII.1 342.
Kelly, G. M. (1989). Elementary observations on 2-categorical limits. Bulletin of the Australian Mathematical Society 39 301317.
Kelly, G. M. (2005). Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Notes Series volume 64, Cambridge University Press, 1982, also available as Repr. Theory Appl. Categ. 10.
Kelly, G. M. and Lack, S. (1993). Finite product-preserving-functors, Kan extensions and strongly-finitary 2-monads. Applied Categorical Structures 1 8594.
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 163179.
Kelly, G. M. and Schmitt, V. (2005). Notes on enriched categories with colimits of some class. Theory and Applications of Categories 14 (17) 399423.
Kozen, D. (1994). A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110 (2) 366390.
Kurz, A. and Velebil, J. (2013). Enriched logical connections. Applied Categorical Structures 21 (4) 349377.
Lack, S. (2002) Codescent objects and coherence. Journal of Pure and Applied Algebra 175 223241.
Lack, S. and Rosický, J. (2011). Notions of Lawvere theory. Applied Categorical Structures 19 (1) 363391.
Lair, C. (1996). Sur le genre d'esquissabilité des catégories modelables (accessibles) possédant les produits de deux. Diagrammes 35 2552.
Lawvere, F. W. (2004). Functorial Semantics of Algebraic Theories , Ph.D. thesis, Columbia University 1963, available as Repr. Theory Appl. Categ. 5 1121.
Linton, F. E. J. (1966). Some aspects of equational categories. In: Proc. Conf. Categ. Alg. La Jolla 1965, Springer 8494.
Mac Lane, S. (1971). Categories for the Working Mathematician, Springer.
Métayer, F. State monads and their algebras. arXiv:math.CT/0407251v1.
Pin, J.-E. (1997). Syntactic semigroups. Chap. 10 In: Handbook of Language Theory, volume I, Springer-Verlag 679746.
Raftery, J. (2013). Order algebraizable logics. Annals of Pure and Applied Logic 164 251283.
Scott, D. (1971). The Lattice of Flow Diagrams LNM volume 188, Springer-Verlag 311366.
Street, R. (1974). Fibrations and Yoneda's Lemma in a 2-Category, Lecture Notes in Mathematics volume 420, Springer 104133.
Street, R. (1982). Two-dimensional sheaf theory. Journal of Pure and Applied Algebra 24 251270.
Street, R. and Walters, R. F. C. (1978). Yoneda structures on 2-categories. Journal of Algebra 50 350379.
Vitale, E. (1994). On the characterization of monadic categories over set. Cahiers de Topologie et Géometrie Differentielle XXXV.4 351358.
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Mathematical Structures in Computer Science
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