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On the structure of equationally complete varieties. II


Author: Don Pigozzi
Journal: Trans. Amer. Math. Soc. 264 (1981), 301-319
MSC: Primary 08B05; Secondary 18C05
DOI: https://doi.org/10.1090/S0002-9947-1981-0603765-1
MathSciNet review: 603765
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Abstract: Each member $ \mathcal{V}$ of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety $ \mathcal{K}$ that satisfies certain very weak versions of the amalgamation and joint embedding properties is isomorphic, as a category, to a coreflective subcategory of some equationally complete subvariety of $ \mathcal{V}$. Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of $ \mathcal{V}$ that fail to have the amalgamation property and fail to be residually small. The family of varieties universal in the above sense includes commutative groupoids, bisemigroups (i.e., algebras with two independent associative operations), and quasi-groups.


References [Enhancements On Off] (What's this?)

  • [1] P. D. Bacsich, Amalgamation properties and interpolation theorems for equational theories, Algebra Universalis 5 (1975), 45-55. MR 0381984 (52:2873)
  • [2] B. Banaschewski, Injectivity and essential extensions in equational classes of algebras, Proc. Conf. Universal Algebra, October 1969, ed. G. H. Wenzel, Queen's Papers in Pure and Appl. Math., No. 25, Queen's University, Kingston, Ontario, 1970, pp. 131-147. MR 0258708 (41:3354)
  • [3] G. Bergman, The diamond lemma in ring theory, Advances in Math. 29 (1978), 178-218. MR 506890 (81b:16001)
  • [4] D. M. Clark and P. H. Krauss, Para primal algebras, Algebra Universalis 6 (1976), 165-192. MR 0422113 (54:10105)
  • [5] S. Fajtlowicz, Problem $ 644$, Colloq. Math. 19 (1969), 334.
  • [6] Z. Hedrlin and J. Lambek, How comprehensive is the category of semigroups?, J. Algebra 11 (1969), 195-212. MR 0237611 (38:5892)
  • [7] Z. Hedrlin and A. Pultr, On full embeddings of categories of algebras, Illinois J. Math. 10 (1966), 392-406. MR 0191858 (33:85)
  • [8] B. Jónsson, Extensions of relational systems, The Theory of Models, Proc. 1963 Internat. Sympos. at Berkeley, Eds. J. W. Addison, L. Henkin and A. Tarski, North-Holland, Amsterdam, 1965, pp. 146-157.
  • [9] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics $ 5$, Springer-Verlag, New York, 1971. MR 0354798 (50:7275)
  • [10] A. I. Mal'cev, The structural characterization of certain classes of algebras, Dokl. Akad. Nauk SSSR 120 (1958), 29-32; English transl., The metamathematics of algebra systems, North-Holland, Amsterdam, 1971, pp. 56-60.
  • [11] -, Algebraic systems, Springer-Verlag, New York, 1973. MR 0349384 (50:1878)
  • [12] G. F. McNulty, The decision problem for equational bases of algebras, Ann. Math. Logic 10 (1976), 193-259. MR 0432440 (55:5428)
  • [13] D. Pigozzi, The universality of the variety of quasigroups, J. Austral. Math. Soc. Ser. A 21 (1976), 194-219. MR 0392769 (52:13582)
  • [14] -, Universal equational theories and varieties of algebras, Ann. of Math. Logic. 17 (1979), 117-150. MR 552418 (81b:03034)
  • [15] -, On the structure of equationally complete varieties. I, Colloq. Math. (to appear). MR 665782 (83j:08005)
  • [16] J. Sichler, Testing categories and strong universality, Canad. J. Math. 25 (1973), 370-385. MR 0318258 (47:6805)
  • [17] W. Taylor, Residually small categories, Algebra Universalis 2 (1972), 33-55. MR 0314726 (47:3278)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0603765-1
Article copyright: © Copyright 1981 American Mathematical Society

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