Abstract
Assuming the usual finite axiom schema of polyadic equality algebras, axiom (P10) is changed to a stronger version. It is proved that infinite dimensional, polyadic equality algebras satisfying the resulting system of axioms are representable. The foregoing stronger axiom is not given with a first order schema. The latter is to be expected knowing the negative results for the Halmos schema axiomatizability of the representable, infinite dimensional, polyadic equality algebras. Furthermore, Halmos’ well-known classical theorem that “locally finite polyadic equality algebras of infinite dimension α are representable” is generalized for locally-\({\mathfrak{m}}\) polyadic equality algebras, where \({\mathfrak{m}}\) is an arbitrary infinite cardinal and \({\mathfrak{m}}\) < α. Also, a neat embedding theorem is proved for the foregoing classes of polyadic-like equality algebras (a neat embedding theorem does not exists for polyadic equality algebras).
Similar content being viewed by others
References
Andréka H.: A finite axiomatization of locally square cylindric-relativized set algebras. Studia Sci. Math. Hungar. 38, 1–11 (2001)
Andréka, H., Ferenczi, M. and Németi, I.: Cylindric-like Algebras and Algebraic Logic. Bolyai Society Mathematical Studies, Springer (2012)
Cirulis J.: Generalizing the notion of polyadic algebra. Bull. Sect. Logic Univ. Łódź, 2–7 (1986)
Daigneault A., Monk J. D.: Representation theory for polyadic algebras. Fund. Math. 52, 151–176 (1963)
Ferenczi M.: On the representability of neatly embeddable CA’s by relativized set algebras. Algebra Universalis 63, 331–350 (2010)
Ferenczi M.: The polyadic generalization of the Boolean axiomatization of fields of sets. Trans. Amer. Math. Soc. 364, 867–886 (2012)
Georgescu G.: A representation theorem for polyadic Heyting algebras. Algebra Universalis 14, 197–209 (1982)
Givant, S. and Halmos, P.: Introduction to Boolean Algebras, Springer (2009)
Halmos P.: Algebraic logic II., Homogeneous, locally finite polyadic Boolean algebras. Fund. Math. 43, 255–325 (1956)
Halmos P.: Algebraic logic IV., Equality in polyadic algebras. Trans. Amer. Math. Soc. 86, 1–27 (1957)
Henkin, L., Monk, J. D., Tarski, A., Andréka, H., Németi, I.: Cylindric Set algebras. Lecture Notes in Math., 883, Springer (1981)
Henkin, L., Monk, J. D. and Tarski, A.: Cylindric Algebras I-II. North Holland (1985)
Hirsch, R. and Hodkinson, I.: Relation Algebras by Games. North Holland (2002)
Jónsson B., Tarski A.: Boolean algebras with operators. J. Symbolic Logic 18, 1–96 (1953)
Keisler H.J.: A complete first order logic with infinitary predicates. Fund. Math. 52, 177–203 (1963)
Németi I., Sági G.: On the equational theory of representable polyadic algebras. J. Symbolic Logic 65, 1143–1167 (2000)
Resek, D.: Some results on relativized cylindric algebras. Ph.D. Thesis, University of California, Berkeley (1975)
Slominski, J.: The theory of abstract algebras with infinitary operations. Rozprawy Mat. 18 (1959)
Sayed Ahmed T.: Some results about neat reducts. Algebra Universalis 63, 17–36 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by I. Hodkinson.
Rights and permissions
About this article
Cite this article
Ferenczi, M. Representations of polyadic-like equality algebras. Algebra Univers. 75, 107–125 (2016). https://doi.org/10.1007/s00012-015-0360-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-015-0360-1