An equational logic sampler

• J. W. Addison [1954] On some points of the theory of recursive functions, Ph.D. Thesis, University of Wisconsin, Madison.

  Google Scholar 

• S. I. Adjan [1958] On algorithmic problems in effectively complete classes of groups (Russian), Doklady Akad. Nauk SSSR23, 13–16.

  Google Scholar 

• K. Baker [1977] Finite equational bases for finite algebras in congruence distributive varieties, Advances in Mathematics24, 207–243.

  Google Scholar 

• K. Baker, G. McNulty, and H. Werner [1987] The finitely based varieties of graph algebras, Acta Scient. Math. (Szeged)51, 3–15.

  Google Scholar 

• K. Baker, G. McNulty, and H. Werner [1989] Shift-automorphism methods for inherently nonfinitely based varieties of algebras, Czechoslovak Math. J. (to appear).

  Google Scholar 

• G. Bauer and F. Otto [1984] Finite complete rewriting systems and the complexity of the word problem, Acta Informatica21, 521–540.

  Google Scholar 

• A. D. Bol'bot [1970] Varieties of Ω-algebras, Algebra i Logika9, 406–415; English Translation Algebra and Logic9, 244–248.

  Google Scholar 

• R. Book [1987] Thue systems as rewriting systems, in Jouannaud [1987], pp. 39–68.

  Google Scholar 

• W. Boone [1954] Certain simple unsolvable problems in group theory, Indag. Math.16, 231–237.

  Google Scholar 

• R. Bryant [1982] The laws of finite pointed groups, Bull. London Math. Soc.14, 119–123.

  Google Scholar 

• B. Buchberger [1987] History and basic features of the critical-pair/completion procedure, in Jouannaud [1987], pp. 3–38.

  Google Scholar 

• S. Burris and E. Nelson [1971a] Embedding the dual of Π _m in the lattice of equational classes of commutative semigroups, Proc. Amer. Math. Soc.30, 37–39.

  Google Scholar 

• [1971b] Embedding the dual of Π_∞ in the lattice of equational classes of semigroups, Algebra Universalis 1, 248–253.

  Google Scholar 

• S. Burris and H. P. Sankappanavar [1981] A Course in Universal Algebra, Springer-Verlag, New York.

  Google Scholar 

• P.M. Cohn [1965] Universal Algebra, Harper & Row, New York.

  Google Scholar 

• P.M. Cohn [1981] Universal Algebra, second edition, D. Reidel, Dordrecht, Holland.

  Google Scholar 

• A. Day [1969] A characterization of modularity of congruence lattices of algebras, Canad. Math. Bull.12, 167–173.

  Google Scholar 

• R. Dedekind [1900] Über die von drei Moduln erzeugte Dualgruppe, Math. Ann.53, 371–403.

  Google Scholar 

• M. Dehn [1911] Über unendlicher diskontinuierliche Gruppen, Math. Ann.71, 116–144.

  Google Scholar 

• N. Dershowitz [1987] Termination of rewriting, in Jouannaud [1987], pp. 69–116.

  Google Scholar 

• T. Evans [1969] Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc.1, 399–403.

  Google Scholar 

• [1978] Word problems, Bull. Amer. Math. Soc. 84, 789–802.

  Google Scholar 

• W. J. Feeney [1954] Certain Unsolvable Problems in the Theory of Cancellation Semigroups, Ph.D. Thesis, Catholic University of America.

  Google Scholar 

• R. Freese [1980] Free modular lattices, Trans. Amer. Math. Soc.261, 81–91.

  Google Scholar 

• R. Freese and O. Garcia, eds. [1983] Universal Algebra and Lattice Theory, Proceedings, Puebla 1982, Lecture Note in Mathematics, vol. 1004, Springer-Verlag, Berlin.

  Google Scholar 

• R. Freese and R. McKenzie [1981] Residually small varieties with modular congruence lattices, Trans. Amer. Math. Soc.264, 419–430.

  Google Scholar 

• [1987] Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Note Series vol. 125, 277pp., Cambridge University Press, Cambridge, England.

  Google Scholar 

• R. Freese, R. McKenzie, G. McNulty, and W. Taylor [1989] Algebras, Lattices, Varieties, Volume II, Wadsworth & Brook/Cole, Monterey, CA (to appear).

  Google Scholar 

• G. Grätzer [1968] Universal Algebra, University Series in Higher Mathematics, D. Van Nostrand Company, New York.

  Google Scholar 

• [1979] Universal Algebra, Second edition of Grätzer [1968], Springer-Verlag, New York, pp. 581.

  Google Scholar 

• R. Gurevič [1989] Equational theory of positive numbers with exponentiation is not finitely axiomatizable, Annals of Pure and Applied Logic (to appear).

  Google Scholar 

• L. Henkin [1977] The logic of equality, Amer. Math. Monthly84, 597–612.

  Google Scholar 

• C. Herrmann [1983] On the word problem for the modular lattice with four free generators, Math. Ann.256, 513–527.

  Google Scholar 

• D. Hobby and R. McKenzie [1988] The Structure of Finite Algebras (Tame Congruence Theory), Contemporary Mathematics, Vol. 76, American Mathematical Society, Providence, RI.

  Google Scholar 

• G. Huet and D. Lankford [1978] On the uniform halting problem for term rewriting systems, Rapport Laboria 283, Institut de Recherche en Informatique eten Automatique, Le Chesnay, France.

  Google Scholar 

• E. Jacobs and R. Schwabauer [1964] The lattice of equational classes of algebras with one unary operation, Amer. Math. Monthly71, 151–155.

  Google Scholar 

• J. Ježek [1969] Primitive classes of algebras with unary and nullary operations, Colloq. Math.20, 159–179.

  Google Scholar 

• [1976] Intervals in the lattice of varieties, Algebra Universalis 6, 147–158.

  Google Scholar 

• [1981] The lattice of equational theories, part I: modular elements, Czechoslovak Math. J. 31, 127–152.

  Google Scholar 

• [1981] The lattice of equational theories part II: the lattice of full sets of terms, Czechoslovak Math. J. 31, 573–603.

  Google Scholar 

• [1982] The lattice of equational theories part III: definability and automorphisms, Czechoslovak Math. J. 32, 129–164.

  Google Scholar 

• [1985] Nonfinitely based three-element groupoids, Algebra Universalis 20, 292–301.

  Google Scholar 

• [1986] The lattice of equational theories part IV: equational theories of finite algebras, Czechoslovak Math. J. 36, 331–341.

  Google Scholar 

• B. Jónsson [1967] Algebras whose congruence lattices are distributive, Math. Scand.21, 110–121.

  Google Scholar 

• [1980] Congruence varieties, Algebra Universalis 10, 355–394.

  Google Scholar 

• J.-P. Jouannaud ed. [1987] Rewriting Techniques and Applications, Academic Press, New York.

  Google Scholar 

• C. Kalfa [1986] Decision problems concerning properties of finite sets of equations, J. Symbolic Logic51, 79–87.

  Google Scholar 

• J. Kalicki [1952] On comparison of finite algebras, Proc. Amer. Math. Soc.3, 36–40.

  Google Scholar 

• [1955] The number of equationally complete classes of equations, Indag. Math. 58, 660–662.

  Google Scholar 

• E. Kiss and P. Pröhle [1988] Problems and results in tame congruence theory, Mathematmatical Institute of the Hungarian Academy of Sciences, preprint no. 60/1988.

  Google Scholar 

• R. Kruse [1973] Identities satisfied by a finite ring, J. Algebra26, 298–318.

  Google Scholar 

• W. Lampe [1987] A property of lattices of equational theories, Algebra Universalis24.

  Google Scholar 

• I. V. L'vov [1973] Varieties of associative rings, I, Algebra i Logika12, 269–297; II, Algebra i Logika12, 667–688, 735.

  Google Scholar 

• R. Lyndon [1951] Identities in two-valued calculi, Trans. Amer. Math. Soc.71, 457–465.

  Google Scholar 

• [1954] Identities in finite algebras, Proc. Amer. Math. Soc. 5, 8–9.

  Google Scholar 

• A. I. Maltsev [1958] On homomorphisms onto finite groups, Ucen. Zap. Ivan. Ped. Inst.18, 49–60; English translation, Amer. Math. Soc. Transl.119, 67–79.

  Google Scholar 

• [1966] Identical relations on varieties of quasigroups, Mat. Sbornik 69, 3–12; English translation, (1969), Amer. Math. Soc. Transl. 82, 225–235; Chapter 29 in Maltsev [1971].

  Google Scholar 

• A. I. Maltsev [1968] Problems on the borderline between algebra and logic, in: Proc. Int. Congress of Math. (Moscow 1966) Moscow, Mir, 1968 pp.217–231; English translation, (1968), Amer. Math. Soc. Transl.70, 89–100; Chapter 34 in Maltsev [1971].

  Google Scholar 

• [1971] The Metamathematics of Algebraic Systems, A selection of Maltsev's papers, edited and translated by B. F. Wells, III, North-Holland Publishing Company, Amsterdam.

  Google Scholar 

• [1973] Algebraic Systems, Akademie-Verlag, Berlin.

  Google Scholar 

• A. Markov [1947] On the impossibility of certain algorithms in the theory of associative systems, Doklady Akad, Nauk SSSR55, 587–590.

  Google Scholar 

• [1951] Impossibility of certain algorithms in the theory of associative systems, Doklady Akad. Nauk SSSR 77, 953–956.

  Google Scholar 

• C. Martin [1973] The Equational Theories of Natural Numbers and Transfinite Ordinals, Ph.D. Thesis, University of California, Berkeley, CA.

  Google Scholar 

• R. McKenzie [1970] Equational bases for lattice theories, Math. Scand.27, 24–38.

  Google Scholar 

• [1971] Definability in lattices of equational theories, Ann. Math. Logic 3, 197–237.

  Google Scholar 

• [1975] On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model, J. Symbolic Logic 40, 186–196.

  Google Scholar 

• [1984] A new product of algebras and a type reduction theorem, Algebra Universalis 18, 29–69.

  Google Scholar 

• R. McKenzie [1988] Finite equational bases for congruence modular algebras, Algebra Universalis (to appear).

  Google Scholar 

• R. McKenzie, G. McNulty, and W. Taylor [1987] Algebras, Lattices, Varieties, Volume I, Wadsworth & Brooks/Cole, Monterey, CA.

  Google Scholar 

• R. McKenzie and M. Valeriote [1989] The Structure of Decidable Locally Finite Varieties, Birkhäuser Verlag, Boston, MA (to appear).

  Google Scholar 

• J. C. C. McKinsey [1943] The decision problem for some classes of sentences without quantifiers, J Symbolic Logic8, 61–76.

  Google Scholar 

• G. McNulty [1976a] The decision problem for equational bases of algebras, Ann. Math. Logic11, 193–259.

  Google Scholar 

• [1976b] Undecidable properties of finite sets of equations, J. Symbolic Logic 41, 589–604.

  Google Scholar 

• [1981] Structural diversity in the lattice of equational theories, Algebra Universalis 13, 271–292.

  Google Scholar 

• [1986] Fifteen possible previews in equational logic, in Szabó and Szendrei [1986].

  Google Scholar 

• G. McNulty and C. Shallon [1983] Inherently nonfinitely based finite algebras, in Freese and Garcia [1983], pp. 205–231.

  Google Scholar 

• A. Mekler, E. Nelson, and S. Shelah [1989] A variety with solvable, but not uniformly solvable, word problem.

  Google Scholar 

• V. L. Murskii [1965] The existence in three-valued logic of a closed class without a finite complete system of identities, Doklady Akad. Nauk SSSR163, 815–818.

  Google Scholar 

• [1968] Examples of varieties of semigroups, Mat. Zametki 3, 663–670.

  Google Scholar 

• [1971] Nondiscernible properties of finite systems of identity relations, Doklady Akad. Nauk SSSR 196, 520–522.

  Google Scholar 

• [1975] The existence of a finite basis and some other properties of “almost all” finite algebras, Problemy Kibernet. 30, 43–56.

  Google Scholar 

• [1979] Concerning the number of k-element algebras with one binary operation which have no finite basis of identities, Problemy Kibernet. 35, 5–27.

  Google Scholar 

• S. Oates and M. B. Powell [1965] Identical relations in finite groups, J. Algebra1, 11–39.

  Google Scholar 

• S. Oates-Williams and M. Vaughan-Lee [1978] Varieties that make one Cross, J. Austral. Math. Soc.(A)26, 368–382.

  Google Scholar 

• C. O'Dúnlaing [1983] Undecidable questions of Thue systems, Theoret. Comp. Sci.23, 339–345.

  Google Scholar 

• P. Perkins [1966] Decision Problems for Equational Theories of Semigroups and General Algebras, Ph.D. Thesis, University of California, Berkeley, CA.

  Google Scholar 

• [1967] Unsolvable problems for equational theories, Notre Dame J. of Formal Logic 8, 175–185.

  Google Scholar 

• [1968] Bases for equational theories of semigroups, J. Algebra 11, 293–314.

  Google Scholar 

• [1984] Basis questions for general algebra, Algebra Universalis 19, 16–23.

  Google Scholar 

• C. S. Peirce [1880] On the algebra of logic, Amer. J. Math.3, 15–57.

  Google Scholar 

• R. S. Pierce [1968] Introduction to the Theory of Abstract Algebras, Holt, Rinehart and Winston, New York.

  Google Scholar 

• D. Pigozzi [1976] Base-undecidable properties of universal varieties, Algebra Universalis6, 193–223.

  Google Scholar 

• D. Pigozzi and G. Tardos [1989] The representation of certain abstract lattices aslattices of subvarieties, (to appear).

  Google Scholar 

• D. Plaisted [1985] The undecidability of self embedding for term rewriting systems, Inf. Proc. Lett.20, 61–64.

  Google Scholar 

• S. V. Polin [1976] On the identities of finite algebras, Sib. Mat. J.17, 1356–1366.

  Google Scholar 

• E. Post [1941] The Two-valued Iterative Systems of Mathematical Logic, Annals of Math. Studies, No. 5, Princeton University Press, Princeton, NJ.

  Google Scholar 

• [1947] Recursive unsolvability of a problem of Thue, J. Symbolic Logic 12, 1–11.

  Google Scholar 

• M. Rabin [1958] Recursive unsolvability of group theoretic problems, Ann. of Math.67, 172–194.

  Google Scholar 

• M. Sapir [1987a] Problems of Burnside type and the finite basis property in varieties of semigroups, Izv. Akad. Nauk SSSR51; English translation, Math. USSR Izvestiya30, 295–314.

  Google Scholar 

• [1987b] Inherently nonfinitely based finite semigroups, Mat. Sbornik 133; English Translation, Math. USSR Sbornik 61 (1988), 155–166.

  Google Scholar 

• E. Schröder [1890] Vorlesungen über die Algebra der Logik (exact Logik), edited in part by E. Müller and B.G. Teubner, published in four volumes, Leipzig; Second edition in three volumes, (1966), Chelsea Publ. Co., New York.

  Google Scholar 

• K. Schütte, ed. [1968] Contributions to Mathematical Logic, North-Holland Publishing Co., Amsterdam.

  Google Scholar 

• D. Scott [1956] Equationally complete extensions of finite algebras, Indag. Math.59, 35–38.

  Google Scholar 

• C. Shallon [1979] Nonfinitely Based, Binary Algebra Derived for Lattices, Ph.D. Thesis, University of California, Los Angeles, CA.

  Google Scholar 

• L. N. Shevrin and M. V. Volkov [1985] Identities of semigroups, Izvestiya VUZ Mat.29, 3–47.

  Google Scholar 

• D. Smith [1972] Non-recursiveness of the set of finite sets of equations whose theories are one based, Notre Dame J. Formal Logic13, 135–138.

  Google Scholar 

• L. Szabó and A. Szendrei, eds. [1986] Lectures in Universal Algebra, Colloquia Mathematica Societas János Bolyai vol. 43 pp. 307–331, North-Holland, Amsterdam.

  Google Scholar 

• A. Szendrei [1986] Clones in Universal Algebra, Séminaire de Mathématiques Supérieures vol. 99, Les Presses de l'Université de Montréal, Montréal, Canada.

  Google Scholar 

• W. Szmielew [1954] Elementary properties of Abelian groups, Fund. Math.41, 203–271.

  Google Scholar 

• A. Tarski [1941] On the calculus of relations, J. Symbolic Logic6, 73–89.

  Google Scholar 

• [1949a] Arithmetical classes and types of Boolean Algebras (preliminary report), Bull. Amer. Math. Soc. 55, p. 64.

  Google Scholar 

• [1949b] Undecidability of the theories of lattices and projective geometries, J. Symbolic Logic 14, 77–78.

  Google Scholar 

• [1953] A formalization of set theory without variables (abstract), J. Symbolic Logic 18, p. 189.

  Google Scholar 

• A. Tarski [1968] Equational logic, in Schütte [1968], pp. 275–288.

  Google Scholar 

• A. Tarski and S. Givant [1987] A Formalization of Set Theory Without Variables, Colloquium Publications, vol 41, Amer. Math. Soc., Providence, RI.

  Google Scholar 

• W. Taylor [1979] Equational logic, Houston J. Math. (Survey Issue).

  Google Scholar 

• [1986] The Clone of a Topological Space, Research and Exposition in Mathematics, vol. 13, Heldermann Verlag, Berlin.

  Google Scholar 

• M. Vaughan-Lee [1979] Laws in finite loops, Algebra Universalis9, 269–280.

  Google Scholar 

• M. Vaughan-Lee [1983] Nilpotence in permutable varieties, in Freese and Garcia [1983], pp. 293–308.

  Google Scholar 

• B. Wells [1982] Pseudorecursive Varieties and Their Word Problems, Ph.D. Thesis, University of California, Berkeley, CA.

  Google Scholar 

Download references