On the structure of equationally complete varieties. II
Author:
Don Pigozzi
Journal:
Trans. Amer. Math. Soc. 264 (1981), 301319
MSC:
Primary 08B05; Secondary 18C05
MathSciNet review:
603765
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Each member of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety 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 . Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of 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 quasigroups.
 [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. on Universal Algebra (Queen’s
Univ., Kingston, Ont., 1969) Queen’s Univ., Kingston, Ont., 1970,
pp. 131–147. MR 0258708
(41 #3354)
 [3]
George
M. Bergman, The diamond lemma for ring theory, Adv. in Math.
29 (1978), no. 2, 178–218. MR 506890
(81b:16001), http://dx.doi.org/10.1016/00018708(78)900105
 [4]
David
M. Clark and Peter
H. Krauss, Para primal algebras, Algebra Universalis
6 (1976), no. 2, 165–192. MR 0422113
(54 #10105)
 [5]
S. Fajtlowicz, Problem , 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
(38 #5892)
 [7]
Z.
Hedrlín 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, NorthHolland, Amsterdam, 1965, pp. 146157.
 [9]
Saunders
MacLane, Categories for the working mathematician,
SpringerVerlag, New YorkBerlin, 1971. Graduate Texts in Mathematics, Vol.
5. MR
0354798 (50 #7275)
 [10]
A. I. Mal'cev, The structural characterization of certain classes of algebras, Dokl. Akad. Nauk SSSR 120 (1958), 2932; English transl., The metamathematics of algebra systems, NorthHolland, Amsterdam, 1971, pp. 5660.
 [11]
A.
I. Mal′cev, Algebraic systems, SpringerVerlag, New
YorkHeidelberg, 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
(50 #1878)
 [12]
George
F. McNulty, The decision problem for equational bases of
algebras, Ann. Math. Logic 10 (1976), no. 34,
193–259. MR 0432440
(55 #5428)
 [13]
Don
Pigozzi, The universality of the variety of quasigroups, J.
Austral. Math. Soc. Ser. A 21 (1976), no. 2,
194–219. MR 0392769
(52 #13582)
 [14]
Don
Pigozzi, Universal equational theories and varieties of
algebras, Ann. Math. Logic 17 (1979), no. 12,
117–150. MR
552418 (81b:03034), http://dx.doi.org/10.1016/00034843(79)900238
 [15]
Don
Pigozzi, On the structure of equationally complete varieties.
I, Colloq. Math. 45 (1981), no. 2,
191–201. MR
665782 (83j:08005)
 [16]
J.
Sichler, Testing categories and strong universality, Canad. J.
Math. 25 (1973), 370–385. MR 0318258
(47 #6805)
 [17]
Walter
Taylor, Residually small varieties, Algebra Universalis
2 (1972), 33–53. MR 0314726
(47 #3278)
 [1]
 P. D. Bacsich, Amalgamation properties and interpolation theorems for equational theories, Algebra Universalis 5 (1975), 4555. 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. 131147. MR 0258708 (41:3354)
 [3]
 G. Bergman, The diamond lemma in ring theory, Advances in Math. 29 (1978), 178218. MR 506890 (81b:16001)
 [4]
 D. M. Clark and P. H. Krauss, Para primal algebras, Algebra Universalis 6 (1976), 165192. MR 0422113 (54:10105)
 [5]
 S. Fajtlowicz, Problem , Colloq. Math. 19 (1969), 334.
 [6]
 Z. Hedrlin and J. Lambek, How comprehensive is the category of semigroups?, J. Algebra 11 (1969), 195212. MR 0237611 (38:5892)
 [7]
 Z. Hedrlin and A. Pultr, On full embeddings of categories of algebras, Illinois J. Math. 10 (1966), 392406. 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, NorthHolland, Amsterdam, 1965, pp. 146157.
 [9]
 S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics , SpringerVerlag, 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), 2932; English transl., The metamathematics of algebra systems, NorthHolland, Amsterdam, 1971, pp. 5660.
 [11]
 , Algebraic systems, SpringerVerlag, New York, 1973. MR 0349384 (50:1878)
 [12]
 G. F. McNulty, The decision problem for equational bases of algebras, Ann. Math. Logic 10 (1976), 193259. MR 0432440 (55:5428)
 [13]
 D. Pigozzi, The universality of the variety of quasigroups, J. Austral. Math. Soc. Ser. A 21 (1976), 194219. MR 0392769 (52:13582)
 [14]
 , Universal equational theories and varieties of algebras, Ann. of Math. Logic. 17 (1979), 117150. 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), 370385. MR 0318258 (47:6805)
 [17]
 W. Taylor, Residually small categories, Algebra Universalis 2 (1972), 3355. MR 0314726 (47:3278)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
08B05,
18C05
Retrieve articles in all journals
with MSC:
08B05,
18C05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198106037651
PII:
S 00029947(1981)06037651
Article copyright:
© Copyright 1981
American Mathematical Society
