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Zero-dimensional compact associative distributive universal algebras

Author: Tae Ho Choe
Journal: Proc. Amer. Math. Soc. 42 (1974), 607-613
MSC: Primary 08A05
MathSciNet review: 0325492
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Abstract: We consider the question under which conditions a zero-dimensional compact universal algebra $ \mathfrak{A}$ of finite type is profinite in the sense that the intersection of all closed congruences on $ \mathfrak{A}$ with finite quotients is trivial. This is known ([2], [9]) to be the case for typical algebras such as groups, semigroups, Boolean lattices (or distributive lattices) and associative rings, but not in general. In this paper we show that if the underlying algebra of the $ \mathfrak{A}$ has generalized associativity and distributivity (see definitions in §1), then $ \mathfrak{A}$ is always profinite. It then follows directly from [2] that the two categories of all residually finite associative, distributive universal algebras of the same finite type and of all zero-dimensional compact ones are in adjoint situation. From this it is shown that all projectives in the latter category are completely characterized in terms of free algebras in the former category.

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  • [1] L. W. Anderson and R. P. Hunter, On residual properties of certain semigroups, Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967), Deutsch. Verlag Wissen., Berlin, 1969, pp. 15-19. MR 39 #7017. MR 0245711 (39:7017)
  • [2] B. Banaschewski, On profinite universal algebras, Proc. Third Topological Conference, Prague, September, 1971. MR 0351960 (50:4448)
  • [3] -, Projective covers in categories of topological spaces and topological algebras, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968), Academia, Prague, 1971, pp. 63-91. MR 44 #1616. MR 0284388 (44:1616)
  • [4] N. Bourbaki, General topology. Part I. Hermann, Paris; Addison-Wesley, Reading Mass., 1966. MR 34 #5044a.
  • [5] G. Grätzer, Universal algebra, Van Nostrand, Princeton, N.J., 1968. MR 40 #1320. MR 0248066 (40:1320)
  • [6] K. W. Gruenberg, Projective profinite groups, J. London Math. Soc. 42 (1967), 155-165. MR 35 #260. MR 0209362 (35:260)
  • [7] S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122.
  • [8] B. Mitchell, Theory of categories, Pure and Appl. Math., vol. 17, Academic Press, New York, 1965. MR 34 #2647. MR 0202787 (34:2647)
  • [9] K. Numakura, Theorems on compact totally disconnected semigroups and lattices, Proc. Amer. Math. Soc. 8 (1957), 623-626. MR 19, 290. MR 0087032 (19:290d)

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Keywords: Universal algebra, compact topological universal algebra, free algebra, projectivity
Article copyright: © Copyright 1974 American Mathematical Society

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