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The undecidability of the elementary theory of lattices of all equational theories of large signature

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The lattice \({\mathcal {L}}_{\Delta }\) of all equational theories of signature \(\Delta \) has an undecidable elementary theory, according to a theorem of Burris and Sankappanavar from 1975, provided \(\Delta \) is large in the sense of providing at least one operation symbol of rank at least two or at least two operation symbols of rank one. On the other hand, Burris also noted in 1971 that the equational theory of \({\mathcal {L}}_{\Delta }\) is decidable. We use the work of Jaroslav Ježek in a effort to find the point along the spectrum from the equational theory to the elementary theory where undecidability enters. We provide four additional proofs that \({\mathcal {L}}_{\Delta }\) has an undecidable elementary theory. Our sharpest result is that the \(\forall ^*\exists ^*\forall ^*\) theory of \({\mathcal {L}}_{\Delta }\) is hereditarily undecidable.

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References

  1. Bernays, P., Schönfinkel, M.: Zum Entscheidungsproblem der mathematischen Logik. Math. Ann. 99(1), 342–372 (1928). https://doi.org/10.1007/BF01459101

    Article  MathSciNet  MATH  Google Scholar 

  2. Burris, S.: On the structure of the lattice of equational classes \({\cal{L}}(\tau )\). Algebra Universalis 1(1), 39–45 (1971). https://doi.org/10.1007/BF02944953

    Article  MathSciNet  MATH  Google Scholar 

  3. Burris, S., McKenzie, R.: Decidable varieties with modular congruence lattices. Bull. Am. Math. Soc. (N.S.) 4(3), 350–352 (1981). https://doi.org/10.1090/S0273-0979-1981-14912-0

    Article  MathSciNet  MATH  Google Scholar 

  4. Burris, S., Sankappanavar, H.P.: Lattice-theoretic decision problems in universal algebra. Algebra Universalis 5(2), 163–177 (1975). https://doi.org/10.1007/BF02485250

    Article  MathSciNet  MATH  Google Scholar 

  5. Eršov, J.L., Lavrov, I.A., Taĭmanov, A.D., Taĭclin, M.A.: Elementary theories. Uspehi Mat. Nauk 20(4(124)), 37–108 (1965)

  6. Eršov, J.L., Taĭclin, M.A.: Undecidability of certain theories. Algebra i Logika Sem. 2(5), 37–41 (1963)

    MathSciNet  Google Scholar 

  7. Freese, R.: Free modular lattices. Trans. Am. Math. Soc. 261(1), 81–91 (1980). https://doi.org/10.2307/1998318

    Article  MathSciNet  MATH  Google Scholar 

  8. Herrmann, C.: On the word problem for the modular lattice with four free generators. Math. Ann. 265(4), 513–527 (1983). https://doi.org/10.1007/BF01455951

    Article  MathSciNet  MATH  Google Scholar 

  9. Jacobs, E., Schwabauer, R.: The lattice of equational classes of algebras with one unary operation. Am. Math. Mon. 71, 151–155 (1964). https://doi.org/10.2307/2311743

    Article  MathSciNet  MATH  Google Scholar 

  10. Ježek, J.: Primitive classes of algebras with unary and nullary operations. Colloq. Math. 20, 159–179 (1969). https://doi.org/10.4064/cm-20-2-159-179

    Article  MathSciNet  MATH  Google Scholar 

  11. Ježek, J.: Intervals in the lattice of varieties. Algebra Universalis 6(2), 147–158 (1976). https://doi.org/10.1007/BF02485826

    Article  MathSciNet  MATH  Google Scholar 

  12. Ježek, J.: The lattice of equational theories. I. Modular elements. Czechoslovak Math. J. 31(106)(1), 127–152 (1981). With a loose Russian summary

    MathSciNet  MATH  Google Scholar 

  13. Ježek, J.: The lattice of equational theories. II. The lattice of full sets of terms. Czechoslovak Math. J. 31(106)(4), 573–603 (1981)

    MathSciNet  MATH  Google Scholar 

  14. Ježek, J.: The lattice of equational theories. III. Definability and automorphisms. Čzechoslovak Math. J. 32(107)(1), 129–164 (1982)

    MathSciNet  MATH  Google Scholar 

  15. Ježek, J.: The lattice of equational theories. IV. Equational theories of finite algebras. Čzechoslovak Math. J. 36(111)(2), 331–341 (1986)

    MathSciNet  MATH  Google Scholar 

  16. Lerman, M.: Degrees of unsolvability. Perspectives in Mathematical Logic. Springer-Verlag, Berlin (1983). https://doi.org/10.1007/978-3-662-21755-9. Local and global theory

    Book  Google Scholar 

  17. Mal’cev, A.I.: Identity relations on manifolds of quasigroups. Mat. Sb. (N.S.) 69(111), 3–12 (1966)

    MathSciNet  Google Scholar 

  18. Markoff, A.: On the impossibility of certain algorithms in the theory of associative systems. C. R. (Doklady) Acad. Sci. URSS (N.S.) 55, 583–586 (1947)

    MathSciNet  MATH  Google Scholar 

  19. McNulty, G.F.: THE DECISION PROBLEM FOR BASES OF EQUATIONAL THEORIES OF ALGEBRAS (1972). Thesis (Ph.D.)–University of California, Berkeley

  20. McNulty, G.F.: The decision problem for equational bases of algebras. Ann. Math. Logic 10(3–4), 193–259 (1976). https://doi.org/10.1016/0003-4843(76)90009-7

    Article  MathSciNet  MATH  Google Scholar 

  21. Monk, J.D.: Mathematical logic. Springer-Verlag, New York-Heidelberg (1976). Graduate Texts in Mathematics, No. 37

  22. Murskiĭ, V.L.: Unrecognizable properties of finite systems of identity relations. Dokl. Akad. Nauk SSSR 196, 520–522 (1971)

    MathSciNet  MATH  Google Scholar 

  23. Perkins, P.: DECISION PROBLEMS FOR EQUATIONAL THEORIES OF SEMIGROUPS AND GENERAL ALGEBRAS (1966). Thesis (Ph.D.)–University of California, Berkeley

  24. Perkins, P.: Unsolvable problems for equational theories. Notre Dame J. Formal Logic 8, 175–185 (1967). http://projecteuclid.org/euclid.ndjfl/1093956081

    Article  MathSciNet  Google Scholar 

  25. Post, E.L.: Recursive unsolvability of a problem of Thue. J. Symb. Logic 12, 1–11 (1947). https://doi.org/10.2307/2267170

    Article  MathSciNet  MATH  Google Scholar 

  26. Rabin, M.O.: A simple method for undecidability proofs and some applications. In: Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.), pp. 58–68. North-Holland, Amsterdam (1965)

  27. Sachs, D.: Identities in finite partition lattices. Proc. Am. Math. Soc. 12, 944–945 (1961). https://doi.org/10.2307/2034397

    Article  MathSciNet  MATH  Google Scholar 

  28. Skolem, T.: Logisch-kombinatorische untersuchungen über die erfüllbarkeit und beweisbarkeit mathematischen sätze nebst einem theoreme über dichte mengen. Videnskapsselskapets skrifter I, Matematisk-natruv. klasse, Videnskabsakademiet i Kristiania 4, 1–36 (1920)

  29. Skolem, T.: Selected works in logic. Edited by Jens Erik Fenstad. Universitetsforlaget, Oslo (1970)

  30. Tarski, A.: Undecidable theories. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam (1953). In collaboration with Andrzej Mostowski and Raphael M. Robinson

  31. Whitman, P.M.: Free lattices. Ann. Math. 2(42), 325–330 (1941). https://doi.org/10.2307/1969001

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    George F. McNulty

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  1. George F. McNulty
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Correspondence to George F. McNulty.

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For the Hawai‘ian Masters Bill Lampe, Ralph Freese, JB Nation (and Jarda Ježek): Mahalo!

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This paper was prepared while the author was supported by the NSF Grant 150016.

This article is part of the topical collection “Algebras and Lattices in Hawaii” edited by W. DeMeo.

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McNulty, G.F. The undecidability of the elementary theory of lattices of all equational theories of large signature. Algebra Univers. 80, 27 (2019). https://doi.org/10.1007/s00012-019-0601-9

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