×

zbMATH — the first resource for mathematics

Characterizing Mal’cev conditions. (English) Zbl 0304.08003

MSC:
08B99 Varieties
08Axx Algebraic structures
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. AczélProof of a theorem on distributive type hyperidentities, Algebra Universalis1 (1971), 1–7. · Zbl 0219.08008
[2] A. A. Akataev and D. M. Smirnov,The lattice of submanifolds of an algebraic manifold (in Russian), Algebra i Logika7, (1968), 5–25; English translation: Algebra and Logic7 (1968), 3–13. · Zbl 0201.00802
[3] A. K. Austin,A note on models of identities, Proc. Amer. Math. Soc.16 (1965), 522–523. · Zbl 0137.00802
[4] A. K. Austin,Finite models for laws in two variables, Proc. Amer. Math. Soc.17 (1966), 1410–1412. · Zbl 0144.01001
[5] J. L. Bell and A. B. Slomson,Models and ultraproducts: an, introduction, North-Holland, Amsterdam, 1969. · Zbl 0179.31402
[6] G. Birkhoff,On the structure of abstract algebras, Proc. Cambridge Philos. Soc.31 (1935), 433–454. · JFM 61.1026.07
[7] G. Birkhoff,Lattice theory (3rd edition), Amer. Math. Soc. Colloq. Publ. #25, Providence, 1967. · Zbl 0153.02501
[8] G. Birkhoff, and O. Frink,Representation of lattices by sets. Trans. Amer. Math. Soc.64 (1948), 299–316. · Zbl 0032.00504
[9] G. Birkhoff and J. D. Lipson,Heterogeneous algebras, J. Combinatorial Theory8 (1970), 115–133. · Zbl 0211.02003
[10] S. Burris,Models in equational theories of unary algebras, Algebra Universalis1 (1972), 386–392. · Zbl 0238.08003
[11] A. Church,Introduction to mathematical logic I, Princeton University Press, Princeton, 1956. · Zbl 0073.24301
[12] P. M. Cohn,Universal algebra, Harper and Row, New York, 1965.
[13] B. Csákány,On the equivalence of certain classes of algebraic systems (in Russian), Acta Sci. Math. (Szeged)23 (1962), 46–57.
[14] B. Csákány,Primitive classes of algebras which are equivalent to classes of semi-modules and modules (in Russian) Acta Sci. Math. (Szeged)24 (1963), 157–164.
[15] B. Csákány,Abelian properties of primitive classes of universal algebras (in Russian), Acta Sci. Math. (Szeged)25, (1964), 202–208.
[16] B. Csákány,Characterization of regular varieties, Acta Sci. Math. (Szeged)31 (1970), 187–189.
[17] A. Day,A characterization of modularity for congruence lattices of algebras, Canad. Math. Bull.12 (1969), 167–173. · Zbl 0181.02302
[18] A. Day,A note on the congruence extension property, Algebra Universalis1 (1971), 234–235. · Zbl 0228.08001
[19] T. Evans,Properties of algebras almost equivalent to, identities, J. London Math. Soc.37 (1962), 53–59. · Zbl 0109.01004
[20] T. Evans,The spectrum of a variety, Z. Math. Logik Grundlagen Math.,13 (1967), 213–218. · Zbl 0199.32703
[21] S. Fajtlowicz,Birkhoff’s theorem in the category of non-indexed algebras, Bull. Acad. Pol. Sci., Sér. sci. math. astr. et phys.17 (1969), 273–275. · Zbl 0179.03602
[22] S. Fajtlowicz,Algebras of homomorphisms, Rend. Math. (6)3 (1970), 523–527. · Zbl 0245.08006
[23] S. Fajtlowicz, W. Holsztyński, J. Mycielski, and B. Weglorz,On powers of bases in some compact algebras, Colloq. Math.19 (1968), 43–46. · Zbl 0156.25802
[24] W. Felscher, Equational maps, 121–161 in H. A. Schmidt, K. Schütte and H.-J. Thiele (eds.)Contributions to mathematical logic, North-Holland, Amsterdam, 1968. · Zbl 0209.04303
[25] K. Fichtner,Varieties of universal algebras with ideals (in Russian), Mat. Sbornik,75 (117) (1968), 445–453; English translation: Math. USSR-Sbornik4 (1968), 411–418.
[26] K. Fichtner,Distributivity and modularity in varieties of algebras, Acta Sci. Math. (Szeged)33 (1972), 343–348. · Zbl 0252.08004
[27] G. A. Fraser and A. Horn,Congruence relations in direct products, Proc. Amer. Math. Soc.26 (1970), 390–394. · Zbl 0241.08004
[28] P. Freyd,Abelian categories, Harper and Row, New York, 1964. · Zbl 0121.02103
[29] N. Funayama and T. Nakayama,On the distributivity of a lattice of lattice-congruences, Proc. Imp. Acad. Tokyo18 (1942), 553–554. · Zbl 0063.01483
[30] A. Goetz,On weak isomorphisms and weak homomorphisms of abstract algebras, Colloq. Math.14 (1966), 163–167. · Zbl 0192.09504
[31] A. Goetz,A generalization of the notion of direct product of universal algebras, Colloq. Math.22 (1971), 167–176. · Zbl 0236.08003
[32] A. Goetz and C. Ryll-Nardzewski,On bases of abstract algebras, Bull. Acad. Pol. Sci., Sér. sci. math., astr. et phys.8 1960), 157–161. · Zbl 0123.00603
[33] G. Grätzer,On spectra of classes of universal algebras, Proc. Amer. Math. Soc.18 (1967), 729–735. · Zbl 0156.25704
[34] G. Grätzer,Universal algebra, van Nostrand, Princeton, 1968.
[35] G. Grätzer,Two Mal’cev-type theorems in universal algebra, J. Combinatorial theory8 (1970), 334–342. · Zbl 0194.01401
[36] G. Grätzer, H. Lakser, and J. Płonka,Joins and direct products of equational classes, Canad. Math. Bull.12 (1969), 741–744. · Zbl 0188.04903
[37] J. Hagemann,On regular and weakly regular congruences, Algebra Universalis (to appear). · Zbl 0273.08001
[38] J. Hagemann and A. Mitschke,On n-permutable congruences Algebra Universalis (to appear).
[39] L. Henkin, J. D. Monk, and A. Tarski,Cylindric algebras, North-Holland, Amsterdam, 1971. · Zbl 0214.01302
[40] P. J. HigginsAlgebras with a scheme of operators, Math. Nachr.27 (1963), 115–132. · Zbl 0117.25903
[41] D. Hilbert,The foundations of geometry, translated by E. J. Townsend, Open Court, Chicago, 1902, 1910. · JFM 33.0082.10
[42] T.-K. Hu,On equational classes of algebras in which congruences of finite direct products are induced by congruences on their factors, mimeographed, Southern Illinois University (1968?).
[43] T.-K. Hu,Stone duality for primal algebra theory, Math. Z.110 (1969), 180–198. · Zbl 0177.02602
[44] T.-K. Hu,Direct factorization in certain classes of universal algebras, Abstract 672–304, Notices Amer. Math. Soc.17 (1970), 169.
[45] T.-K. Hu,On the topological duality for primal algebra theory, Algebra Universalis1 (1971), 152–154. · Zbl 0236.08005
[46] T.-K. Hu and P. Kelenson,Independence and direct factorization of universal algebras, Math. Nachr.51 (1971), 83–99. · Zbl 0223.08004
[47] B. Jónsson,Algebras whose congruence lattices are distributive, Math. Scand.21 (1967), 110–121. · Zbl 0167.28401
[48] L. Kalmár,Contributions to the reduction theory of the decision problem, Fourth paper:Reduction to a finite set of individuals Acta Math. Acad. Sci. Hung.2 (1951), 125–142. · Zbl 0045.00202
[49] D. KellyBasic equations: word problems and Mal’cev conditions, Abstract 701-08-4, Notices Amer. Math. Soc.20 (1973), A-54.
[50] G. Kreisel and J. L. Krivine,Elements of mathematical logic (model theory), North-Holland, Amsterdam, 1967. · Zbl 0155.33801
[51] J.-L. Lagrange,Réflexions sur la résolution algébrique des équations, Nouveaux Mémoires de I’Académie Royale des Sciences et Belles Lettres de Berlin, 1770–1771.
[52] F. W. Lawvere,Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci.50 (1963), 869–872. · Zbl 0119.25901
[53] F. W. Lawvere,Some algebraic problems in the context of functorial semantics of algebraic theories, pp. 51–61 in Lecture Notes in Mathematics61 Springer-Verlag, Berlin, 1968. · Zbl 0204.33802
[54] F. E. J. Linton,Some aspects of equational, categories, pp. 84–89 in: S. Eilenberg, D. K. Harrison, S. MacLane and H. Röhrl, eds.. Proceedings of the conference on categorical algebra, La Jolla, 1965. Springer-Verlag, Berlin and New York, 1966.
[55] J. Łoś,Direct sums in general algebra, Colloq. Math.14 (1966), 33–38. · Zbl 0197.29102
[56] S., MacLane,Categorical algebra, Bull. Amer. Math. Soc.71 (1965), 40–106. · Zbl 0161.01601
[57] A. I. Mal’cev,On the general theory of algebraic systems (in Russian). Math. Sbornik (N. S.)35 (77) (1954), 3–20. English translation: Amer. Math Soc. Translations (2)27 (1963), 125–142.
[58] A. I. Mal’cev,The structural characterization of certain classes of algebras (in Russian), Dokl. Akad. Nauk S.S.S.R.120 (1958), 29–32. English translation: Chapter 9 of [60].
[59] A. I. Mal’cev,Model correspondences, Izv. Akad. Nauk S.S.R. Ser. Mat.23 (1959), 313–316. English translation: Chapter 11 of [60].
[60] A. I. Mal’cev,The metamathematics of algebraic systems, collected papers 1936–67, translated and edited by B.F. Wells III, North-Holland, Amsterdam, 1971.
[61] E. Marczewski,Independence in abstract algebras-results and problems Colloq. Math.14 (1966), 169–188. · Zbl 0192.09502
[62] G. F. McNulty,The decision problem for equational bases of algebras, Ph.D. Thesis, University of California, Berkeley, 1972.
[63] A. Mitschke,Implication algebras are 3-permutable and 3-distributive, Algebra Universalis1 (1971), 182–186. · Zbl 0242.08005
[64] J. B. Nation,Varieties whose congruences satisfy certain lattice identities, Algebra Universalis (to appear). · Zbl 0299.08002
[65] B. H. Neumann,On a problem of G. Grätzer, Publ. Math. Debrecen.,14 (1967), 325–329. · Zbl 0189.29804
[66] B. H. Neumann and E. C. Wiegold,A semigroup representation of varieties of algebras, Colloq. Math.14 (1966), 111–114. · Zbl 0192.09602
[67] W. D. Neumann,Representing varieties of algebras by algebras, J. Austral. Math. Soc.11 (1970), 1–8. · Zbl 0199.32702
[68] R. Padmanabhan,Characterization of a class of groupoids, Algebra Universalis1 (1972), 374–382. · Zbl 0236.20043
[69] B. Pareigis,Categories and functors, Academic Press, New York, 1970. · Zbl 0211.32402
[70] P. Perkins,Unsolvable problems for equational theories, Notre Dame J. Formal Logic3 (1967), 175–185. · Zbl 0197.28201
[71] D. Pigozzi,Amalgamation, congruence-extension and interpolation properties in algebras, Algebra Universalis1 (1972), 269–349. · Zbl 0236.02047
[72] A. F. Pixley,Distributivity and permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc.14 (1963), 105–109. · Zbl 0113.24804
[73] A. F. Pixley,Local Mal’cev conditions, Canad. Math. Bull.15 (1972), 559–568. · Zbl 0254.08009
[74] R. W. Quackenbush,Demi-semi-primal algebras and Mal’cev-type conditions, Math. Z.122 (1971), 166–176. · Zbl 0214.03102
[75] R. W. Quackenbush,Models for Mal’cev conditions: n-permutability, Abstract 71T-A182, Notices Amer. Math. Soc.18 (1971), 806.
[76] P. C. Rosenbloom,Post algebras. I. Postulates and general theory, Amer. J. Math.64 (1942), 167–188. · Zbl 0060.06701
[77] R. Roth,El origen de la teoría de grupos. El teorema de Lagrange (1771), Bol. Math.,3 (1969), 137–141.
[78] D. Sachs,Identities in finite partition lattices, Proc. Amer. Math. Soc.12 (1969), 944–945. · Zbl 0101.02201
[79] B. M. Schein,A remark on bisimple inverse semigroups, Semigroup Forum3 (1971), 80–83. · Zbl 0237.20057
[80] A. Schmidt,Über deduktive Theorien mit mehreren Sorten von Grunddingen, Math. Ann.115 (1938), 485–506 · JFM 64.0028.01
[81] A. Schmidt,Die Zulässigkeit der Behandlung mehrsortiger Theorien mittels der üblichen einsortigen Prädikatenlogik, Math. Ann.123 (1951), 187–200. · Zbl 0042.00605
[82] E. T. Schmidt,Kongruenzrelationen algebraischer Strukturen, Math. Forschungsberichte25, Berlin, 1969.
[83] E. T. Schmidt,Über reguläre Mannigfaltigkeiten, Acta Sci. Math. (Szeged)31 (1970), (197–201. · Zbl 0205.31902
[84] E. T. Schmidt,On n-permutable equational classes, Acta Sci. Math. (Szeged) 33 (1972), 29–30. · Zbl 0253.08002
[85] J. Słomiński,On the determining of the form of congruences in abstract algebras with equationally definable constant elements, Fund. Math.48 (1960), 325–341. · Zbl 0123.00601
[86] S. K. Stein,Finite models of identities, Proc. Amer. Math. Soc.14 (1963), 216–222. · Zbl 0113.24805
[87] M. G. Stone,Proper congruences do not imply a modular congruence lattice, Colloq. Math.23 (1971), 25–27. · Zbl 0275.08004
[88] S. Świerczkowski,On ismorphic free algebras, Fund. Math. 50 (1961) 35–44. · Zbl 0104.25601
[89] A. Tarski,Equational logic and equational theories of algebras, p. 275–288, in: H.A. Schmidt, K. Schütte, and H.-J. Thiele (eds.), Contributions to mathematical logic, North-Holland, Amsterdam, 1968. · Zbl 0209.01402
[90] W. Taylor,Some constructions of compact algebras, Ann. Math. Logic3 (1971), 395–435. · Zbl 0239.08003
[91] W. Taylor,Fixed points of endomorphisms, Algebra Universalis2 (1972), 74–76. · Zbl 0263.08004
[92] W. Taylor,Characterizing Mal’cev conditions, Abstract 72T-A98, Notices Amer. Math. Soc.19 (1972), A431.
[93] W. Taylor,Varieties without doubleton algebras, Abstract 72T-A280, Notices Amer. Math. Soc.19 (1972), A-753.
[94] H. A. Thurston,Derived operations and congruences, Proc. London Math. Soc. (3)8 (1958), 127–134. · Zbl 0078.01901
[95] B. A. TrakhtenbrotThe impossibility of an algorithm for the problem of satisfiability in finite classes, Dokl. Akad. Nauk S.S.S.R. (N.S.)70 (1950), 569–572.
[96] I. I. Valutse,Universal algebras with regular but not permutable congruences (in Russian), Uspekh. Mat. Nauk18 (3) (1963), 145–148.
[97] L. I. Wade,Post algebras and rings, Duke Math. J.12 (1945), 389–395. · Zbl 0063.08105
[98] H. Wang,Logic of many-sorted theories, J. Symbolic Logic,17 (1952), 105–116. · Zbl 0049.14802
[99] R. Wille,Kongruenzklassengeometrien, Lecture notes in mathematics #113, Springer-Verlag, Berlin, 1970.
[100] G. C. Wraith,Algebraic theories, Aarhus University Lecture Notes Series No. 22, April, 1970.
[101] G. C. Wraith,Algebras over theories, Colloq. Math.23 (1971), 181–190. · Zbl 0226.18003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.