Abstract
Partially ordered sets are described in terms of partial operations with equationally defined domains and equations, thus the categoryPOS of posets is represented as a one-sorted essentially algebraic category in the sense of Freyd [7] which, in this case even can be fully embedded into a non-trivial variety. This is achieved by using the relation of a poset rather than its underlying set as the carrier set of the algebraic structure. Essentially equational descriptions of somePOS-based algebraic structures are given, and an equational characterization of Galois connections is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Adámek, H. Herrlich, and J. Rosický: Essentially equational categories,Cahiers Topol. Géom. Différentielles Catégoriques XXIX (1988), 175–192.
J. Adámek, H. Herrlich, and G.E. Strecker:Abstract and Concrete Categories, Wiley Interscience, New York, 1990.
J. Adámek, H. Herrlich, and W. Tholen: Monadic decompositions,J. Pure Appl. Algebra 59 (1989), 111–123.
M. Barr: HSP type theorems in the category of posets, in:Mathematical Foundations of Programming Semantics, LNCS598, Springer, Berlin - New York 1992, 221–234.
M. Barr and C. Wells:Toposes, Triples and Theories, Springer, Berlin - New York 1985.
R. Börger and W. Tholen: Strong, regular and dense generators,Cahiers Topol. Géom. Différentielles Catégoriques XXXII (1991), 257–276.
P. Freyd: Aspects of topoi,Bull. Austral. Math. Soc. 7 (1972), 1–76.
P. Gabriel and F. Ulmer:Lokal präsentierbare Kategorien, LNM221, Springer, Berlin - New York 1971.
H. Herrlich, Regular categories and regular functors,Canad. J. Math. XXVI (1974), 709–720
H. Herrlich, Essentially algebraic categories,Quaestiones Math. 9 (1986) 245–262
S. Mac Lane,Categories for the Working Mathematician, GTM5, Springer, Berlin - New York 1971
E. Makai jun., Automorphisms and Full Embeddings of Categories in Algebra and Topology, in:Category Theory at Work, Heldermann-Verlag, Berlin 1991, 217–260
E.G. Manes,Algebraic Theories, GTM26, Springer, Berlin - New York 1976
L.D. Nel, Initially structured categories,Can. J. Math. XXVII (1975), 1361–1377
H.-E. Porst, T-regular functors, in:Categorical Topology, Heldermann-Verlag, Berlin 1984, 425–440
H.-E. Porst, What is concrete equivalence?Seminarberichte Mathematik 44, Fernuniversität Hagen (1992), 312–321
H.-E. Porst, The Linton Theorem revisited,Cahiers Topol. Géom. Différentielles Catégoriques XXXIV (1993), 229–238.
H. Reichel,Structural Induction on Partial Algebras, Akademie Verlag, Berlin 1984
M. Sekanina, Realisation of ordered sets by means of universal algebras, especially by semigroups, in:Theory of Sets and Topology (in honour of F. Hausdorff), VEB Deutsch. Verlag Wiss. Berlin, 1972, 455–466
Author information
Authors and Affiliations
Additional information
The author gratefully acknowledges the hospitality of the Department of Mathematics, Applied Mathematics and Astronomy at UNISA, where this note was written during an extended visit.
Rights and permissions
About this article
Cite this article
Porst, HE. The algebraic theory of order. Appl Categor Struct 1, 423–440 (1993). https://doi.org/10.1007/BF00872943
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00872943