Abstract
This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class \({\mathcal{A} = \mathbb{ISP}(\mathcal{M})}\), where \({\mathcal{M}}\) is a set, not necessarily finite, of finite algebras, it is shown that each \({{\bf A} \in \mathcal{A}}\) embeds as a topologically dense subalgebra of a topological algebra \({n_{\mathcal{A}}({\bf A})}\) (its natural extension), and that \({n_{\mathcal{A}}({\bf A})}\) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that \({\mathcal{M}}\) is finite and \({\mathcal{A}}\) possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baldwin J.T.: The number of subdirectly irreducible algebras in a variety II. Algebra Universalis 11, 1–6 (1980)
Baldwin J.T., Berman J.: The number of subdirectly irreducible algebras in a variety. Algebra Universalis 5, 379–389 (1975)
Banaschewski, B.: On profinite universal algebras. In: Proc. Third Prague Topological Sympos. (Prague, 1971), 51–62. Academia, Prague (1972)
Bezhanishvili G.: Locally finite varieties. Algebra Universalis 46, 531–548 (2001)
Bezhanishvili G., Grigolia R.: Locally finite varieties of Heyting algebras. Algebra Universalis 54, 465–473 (2005)
Blok, W., Pigozzi, D.: On the structure of varieties with equationally definable principal congruences, I, II and III. Algebra Universalis 15, 195–227 (1982), 18, 334–379 (1984) and 32, 545–608 (1994)
Bonato, A.C.J.: On Residually Small Varieties. Masters’ thesis, University of Waterloo, Ontario, Canada (1994)
Burris, S.N., Sankappanavar, H.P.: A course in universal algebra. Graduate texts in Mathematics, vol. 78. Springer-Verlag (1981) Free download at http://www.math.waterloo.ca/~snburris
Clark D.M., Davey B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998)
Clark D.M., Davey B.A., Freese R.S., Jackson M.: Standard topological algebras: syntactic and principal congruences and profiniteness. Algebra Universalis 52, 343–376 (2004)
Clark D.M., Davey B.A., Jackson M., Pitkethly J.G.: The axiomatizability of topological prevarieties. Adv. Math. 218, 1604–1653 (2008)
Clark D.M., Davey B.A., Pitkethly J.G., Rifqui D.L.: Flat unars: the primal, the semi-primal and the dualisable. Algebra Universalis 63, 303–329 (2010)
Davey B.A.: On the lattice of subvarieties. Houston J. Math. 5, 183–192 (1979)
Davey B.A.: Monotone clones and congruence modularity. Order 6, 389–400 (1979)
Davey, B.A., Gouveia, M.J., Haviar, M., Priestley, H.A.: Multisorted dualisability: change of base. Algebra Universalis (to appear)
Davey B.A., Heindorf L., McKenzie R.: Near unanimity: an obstacle to general duality theory. Algebra Universalis 33, 428–439 (1995)
Davey B.A., Haviar M., Priestley H.A.: The syntax and semantics of entailment in duality theory. J. Symbolic Logic 60, 1087–1114 (1995)
Davey B.A., Haviar M., Priestley H.A.: Boolean topological distributive lattices and canonical extensions. Appl. Categ. Structures 15, 225–241 (2007)
Davey B.A., Jackson M., Pitkethly J.G., Talukder M.R.: Natural dualities for semilattice-based algebras. Algebra Universalis 57, 463–490 (2007)
Davey B.A., Pitkethly J.G.: Dualisability of p-semilattices. Algebra Universalis 45, 149–153 (2001)
Davey B.A., Pitkethly J.G., Willard R.: Dualisability versus residual character: a theorem and a counterexample. Annals Pure Appl. Algebra 210, 423–435 (2007)
Davey B.A., Priestley H.A.: Generalized piggyback dualities and applications to Ockham algebras. Houston J. Math. 13, 151–197 (1987)
Davey B.A., Quackenbush R.W., Schweigert D.: Monotone clones and the varieties they determine. Order 7, 145–167 (1990)
Davey, B.A., Werner, H.: Dualities and equivalences for varieties of algebras. In: Contributions to Lattice Theory Szeged, 1980 (A.P. Huhn and E.T. Schmidt, eds). Coll. Math. Soc. János Bolyai 33, pp.101–275. North-Holland, Amsterdam (1983)
Di Nola A., Niederkorn P.: Natural dualities for varieties of BL-algebras. Arch. Math. Logic 44, 995–1007 (2005)
Dzobiak W.: On infinite subdirectly irreducible algebras in locally finite equational classes. Algebra Universalis 13, 393–304 (1981)
R. Engelking, General Topology. Mathematical Monographs 60, PWN, Warsaw (1977)
Freese R., McKenzie R.: Residually small varieties with modular congruence lattices. Trans. Amer. Math. Soc. 264, 419–430 (1981)
Freese R., Nation J.B.: Congruence lattices of semilattices. Pacific J. Math. 49, 51–58 (1973)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: an Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier (2007)
Gehrke M., Harding J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001)
Gehrke M., Jónsson B.: Bounded distributive lattice expansions. Math. Scand. 94, 13–45 (2004)
Golubov E.A., Sapir M.V.: Varieties of finitely approximable semigroups. Soviet Math. Dokl. 20, 828–832 (1979) (Russian)
Gouveia M.J: A note on profinite completions and canonical extensions. Algebra Universalis 64, 21–24 (2010)
Grätzer, G.: Universal Algebra, 2nd edition. Springer, (1979)
Harding J.: On profinite completions and canonical extensions. Algebra Universalis (Special issue dedicated to Walter Taylor) 55, 293–296 (2006)
Haviar M., Konôpka P., Priestley H.A., Wegener C.B.: Finitely generated free modular ortholattices I. Internat. J. Mathematical Physics 36, 2639–2660 (1997)
Hobby, D., McKenzie, R.: The Structure of Finite Algebras. Contemporary Mathematics, 76 American Mathematical Society (1988) Free download at http://www.ams.org/online_bks/conm76/.
Jackson M.: Dualisability of finite semigroups. Int. J. Algebra Comput. 13, 481–497 (2003)
Kaarli K.: Locally affine complete varieties. J. Austral. Math. Soc. (Series A) 62, 141–159 (1997)
Kaarli K., Pixley A.: Affine complete varieties. Algebra Universalis 24, 74–90 (1987)
Kaarli K., McKenzie R.: Affine complete varieties are congruence distributive. Algebra Universalis 38, 329–354 (1997)
Kaarli K., Pixley A.F.: Polynomial completeness in algebraic systems. Chapman&Hall/CRC, Boca Raton, FL (2001)
Kearnes K.A.: Cardinality bounds for subdirectly irreducible algebras. J. Pure Appl. Algebra 112, 293–312 (1996)
Kearnes, K.A., Kiss, E.A.: The Shape of Congruence Lattices. Mem. Amer. Math. Soc. (to appear) http://spot.colorado.edu/~kearnes/Papers/cong.pdf
Kearnes K.A., Willard R.: Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound. Proc. Amer. Math. Soc. 127, 2841–2850 (1995)
Kearnes, K.A., Willard, R.: Residually finite varieties. In: Conference abstracts, AAA76 (Linz, Austria, 2008)
Kowalski T.: Semisimplicity, EDPC and discriminator varieties of residuated lattices. Studia Logica 77, 255–265 (2004)
Kowalski, T., Kracht, M.: Semisimple varieties of modal algebras Studia Logica 83, 351–363 (2006)
Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, 5, Second Edition Springer (1998)
Maksimova L.: Pretabular superintuitionistic logics. Algebra and Logic 11, 308–314 (1972)
McKenzie R.: Para-primal varieties: a study of finite axiomatizability and definable principal congruences in locally finite varieties. Algebra Universalis 8, 336–348 (1978)
McKenzie R.: Residually small varieties of semigroups. Algebra Universalis 13, 171–201 (1981)
McKenzie R.: Monotone clones, residual smallness and congruence distributivity. Bull. Austral. Math. Soc. 41, 283–300 (1990)
McKenzie R.: The residual bounds of finite algebras. Internat. J. Algebra Comput. 6, 1–28 (1996)
Mitschke, A.: Implication algebras are 3-permutable and 3-distributive. Algebra Universalis 1, 182–186 (1971/1972)
Mitschke A.: Near unanimity identities and congruence distributivity in equational classes. Algebra universalis 8, 29–32 (1978)
Nickodemus, M.H.: An extension of Pontryagin duality for discrete groups. (submitted)
Niederkorn P.: Natural dualities for varieties of MV algebras I. J. Math. Anal. Appl. 255, 58–73 (2001)
Numakura K.: Theorems on compact totally disconnected semigroups and lattices. Proc. Amer. Math. Soc. 8, 623–626 (1957)
Ol’šanskiĭ A. Ju.: Varieties of finitely approximable groups. Izv. Akad. Nauk SSSR Ser. Mat. 33, 915–927 (1969)
Papert D.: Congruences in semi-lattices. J. London Math. Soc. 39, 723–729 (1964)
Pitkethly, J.G., Davey, B.A.: Dualisability: Unary Algebras and Beyond. Advances in Mathematics 9, Springer (2005)
Pixley, A.F.: A survey of interpolation in universal algebra. In: Universal Algebra Colloq. Math. Soc. J. Bolyai, 29, pp. 583–607 (1982)
Priestley, H.A.: Natural dualities for varieties of distributive lattices with a quantifier. In: Algebraic Methods in Logic and in Computer Science (Warsaw, 1991), pp. 291–310 Banach Center Publ., vol. 28, Polish Acd. Sci., Warsaw (1993)
Quackenbush, R.W.: Equational classes generated by finite algebras. Algebra Universalis 1, 265–266 (1971/2)
Quackenbush R., Szabó Cs.: Strong duality for metacyclic groups. J. Austral. Math. Soc. 73, 377–392 (2002)
Ribes, L., Zalesskii, P.: Profinite groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] Springer-Verlag, Berlin (2000)
Sankapannavar H. P.: Principal congruences of pseudocomplemented semilattices and congruence extension property. Proc. Amer. Math. Soc. 73, 308–312 (1979)
Schmid J.: Lee classes and sentences for pseudocomplemented semilattices. Algebra Universalis 25, 223–232 (1988)
Taylor W.: Residually small varieties. Algebra Universalis 2, 33–53 (1972)
Vosmaer, J.: Connecting the profinite completion and the canonical extension using duality. Master of Logic Thesis, Universiteit van Amsterdam (2006) http://www.illc.uva.nl/Publications
Werner, H.: Discriminator Algebras. Studien zur Algebra und Ihre Anwendungen. Band 6. Akademie-Verlag, Berlin (1978)
Willard, R.: An overview of modern universal algebra. In: Logic Colloquium 2004, A. Andretta, K. Kearnes and C. Zambella (eds) Lecture Notes in Logic, Vol. 29, pp. 197–220. Cambridge Univ. Press (2008)
Zádori L.: Monotone Jónsson operations and near unanimity functions. Algebra Universalis 33, 216–236 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by K. Kearnes.
The second author acknowledges support from Portuguese Project ISFL-1-143 of CAUL financed by FCT and FEDER, the third author acknowledges support from Slovak grants APVV-51-009605 and VEGA 1/0485/09.
Rights and permissions
About this article
Cite this article
Davey, B.A., Gouveia, M.J., Haviar, M. et al. Natural extensions and profinite completions of algebras. Algebra Univers. 66, 205 (2011). https://doi.org/10.1007/s00012-011-0155-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-011-0155-y