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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
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.

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  • [1] P. D. Bacsich, Amalgamation properties and interpolation theorems for equational theories, Algebra Universalis 5 (1975), 45–55. MR 0381984
  • [2] B. Banaschewski, Injectivity and essential extensions in equational classes of algebras., Proc. Conf. on Universal Algebra (Queen’s Univ., Kingston, Ont., 1969) Queen’s Univ., Kingston, Ont., 1970, pp. 131–147. MR 0258708
  • [3] George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218. MR 506890, 10.1016/0001-8708(78)90010-5
  • [4] David M. Clark and Peter H. Krauss, Para primal algebras, Algebra Universalis 6 (1976), no. 2, 165–192. MR 0422113
  • [5] S. Fajtlowicz, Problem $ 644$, Colloq. Math. 19 (1969), 334.
  • [6] Zdeněk Hedrlín and Joachim Lambek, How comprehensive is the category of semigroups?, J. Algebra 11 (1969), 195–212. MR 0237611
  • [7] Z. Hedrlín and A. Pultr, On full embeddings of categories of algebras, Illinois J. Math. 10 (1966), 392–406. MR 0191858
  • [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] Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
  • [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] A. I. Mal′cev, Algebraic systems, Springer-Verlag, New York-Heidelberg, 1973. Posthumous edition, edited by D. Smirnov and M. Taĭclin; Translated from the Russian by B. D. Seckler and A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 192. MR 0349384
  • [12] George F. McNulty, The decision problem for equational bases of algebras, Ann. Math. Logic 10 (1976), no. 3-4, 193–259. MR 0432440
  • [13] Don Pigozzi, The universality of the variety of quasigroups, J. Austral. Math. Soc. Ser. A 21 (1976), no. 2, 194–219. MR 0392769
  • [14] Don Pigozzi, Universal equational theories and varieties of algebras, Ann. Math. Logic 17 (1979), no. 1-2, 117–150. MR 552418, 10.1016/0003-4843(79)90023-8
  • [15] Don Pigozzi, On the structure of equationally complete varieties. I, Colloq. Math. 45 (1981), no. 2, 191–201. MR 665782
  • [16] J. Sichler, Testing categories and strong universality, Canad. J. Math. 25 (1973), 370–385. MR 0318258
  • [17] Walter Taylor, Residually small varieties, Algebra Universalis 2 (1972), 33–53. MR 0314726

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Article copyright: © Copyright 1981 American Mathematical Society