Transactions of the American Mathematical Society

Author(s): J. G. Raftery
Journal: Trans. Amer. Math. Soc. 359 (2007), 4405-4427.
MSC (2000): Primary 03B47, 03G25, 06D99, 06F05, 08A50, 08C15
Posted: March 20, 2007
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Abstract: It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The

-generated subdirectly irreducible algebras in this variety are shown to have at most

elements each. A constructive characterization of the subdirectly irreducible algebras is provided, with some applications. The main result implies that every finitely based extension of positive relevance logic containing the mingle and Gödel-Dummett axioms has a solvable deducibility problem.

1.: P. Aglianò, Ternary deductive terms in residuated structures, Acta Sci. Math. (Szeged) 64 (1998), 397-429. MR 1666002 (2000a:08005)
2.: A.R. Anderson, N.D. Belnap, Jr., ``Entailment: The Logic of Relevance and Necessity, Volume 1'', Princeton University Press, 1975. MR 0406756 (53:10542)
3.: A.R. Anderson, N.D. Belnap, Jr., J.M. Dunn, ``Entailment: The Logic of Relevance and Necessity, Volume 2'', Princeton University Press, 1992. MR 1223997 (94b:03042)
4.: A. Avron, The semantics and proof theory of linear logic, Theoretical Computer Science 57 (1988), 161-184. MR 0960102 (90k:68100)
5.: R. Balbes, P. Dwinger, ``Distributive Lattices'', University of Missouri Press, Columbia, 1974. MR 0373985 (51:10185)
6.: W.J. Blok, W. Dziobiak, On the lattice of quasivarieties of Sugihara algebras, Studia Logica 45 (1986), 275-280. MR 0877316 (88d:08005)
7.: W.J. Blok, I.M.A. Ferreirim, On the structure of hoops, Algebra Universalis 43 (2000), 233-257. MR 1774741 (2001g:03120)
8.: W.J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences I, Algebra Universalis 15 (1982), 195-227. MR 0686803 (85g:08005)
9.: W.J. Blok, D. Pigozzi, ``Algebraizable Logics'', Memoirs of the American Mathematical Society, Number 396, Amer. Math. Soc., Providence, 1989. MR 0973361 (90d:03140)
10.: W.J. Blok, J.G. Raftery, Constructing simple residuated lattices, Algebra Universalis 50 (2003), 385-389. MR 2055065 (2005a:06017)
11.: W.J. Blok, J.G. Raftery, Fragments of R-mingle, Studia Logica 78 (2004), 59-106. MR 2108021 (2005i:03027)
12.: W.J. Blok, C.J. van Alten, The finite embeddability property for residuated lattices, pocrims and BCK-algebras, Algebra Universalis 48 (2002), 253-271. MR 1954775 (2003j:06017)
13.: W.J. Blok, C.J. van Alten, On the finite embeddability property for residuated ordered groupoids, Trans. Amer. Math. Soc. 357 (2005), 4141-4157. MR 2159703
14.: S. Burris, H.P. Sankappanavar, ``A Course in Universal Algebra'', Graduate Texts in Mathematics, Springer-Verlag, New York, 1981. MR 0648287 (83k:08001)
15.: C.C. Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc. 88 (1958), 467-490. MR 0094302 (20:821)
16.: M. Dummett, A propositional calculus with a denumerable matrix, J. Symbolic Logic 24 (1959), 97-106. MR 0123476 (23:A801)
17.: J.M. Dunn, Algebraic completeness results for R-mingle and its extensions, J. Symbolic Logic 35 (1970), 1-13. MR 0288008 (44:5206)
18.: K. Dyrda, None of the variety $E_n,\,n\geq 2,$ is locally finite, Demonstratio Math. 20 (1987), 215-219. MR 0941416 (89b:06007)
19.: N. Galatos, ``Varieties of Residuated Lattices'', Ph.D. Thesis, Vanderbilt University, 2003.
20.: N. Galatos, Equational bases for joins of residuated-lattice varieties, Studia Logica 76 (2004), 227-240. MR 2072984 (2005f:06019)
21.: N. Galatos, Minimal varieties of residuated lattices, Algebra Universalis 52 (2005), 215-239. MR 2161651 (2006f:06010)
22.: N. Galatos, H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica, 83 (2006), 279-308. MR 2250112
23.: N. Galatos, J.G. Raftery, Adding involution to residuated structures, Studia Logica 77 (2004), 181-207. MR 2080238 (2005f:06020)
24.: J. Hart, L. Rafter, C. Tsinakis, The structure of commutative residuated lattices, Internat. J. Algebra Comput. 12 (2002), 509-524. MR 1919685 (2003g:06008)
25.: P. Jipsen, C. Tsinakis, A survey of residuated lattices, in ``Ordered Algebraic Structures'', edited by J. Martinez, Kluwer, Dordrecht, 2002, pp.19-56. MR 2083033 (2005e:06024)
26.: B. Jónsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121. MR 0237402 (38:5689)
27.: J. $\L$ os, R. Suszko, Remarks on sentential logics, Proc. Kon. Nederl. Akad. van Wetenschappen, Series A 61 (1958), 177-183. MR 0098670 (20:5125)
28.: A.I. Mal'cev, ``Algebraic Systems'', Springer-Verlag, Berlin, 1973. MR 0349384 (50:1878)
29.: R.K. Meyer, A characteristic matrix for RM, unpublished manuscript, 1967. Partly subsumed in [2, Sec. 29.3].
30.: R.K. Meyer, Conservative extension in relevant implication, Studia Logica 31 (1972), 39-46. MR 0381937 (52:2826)
31.: R.K. Meyer, On conserving positive logics, Notre Dame J. Formal Logic 14 (1973), 224-236. MR 0327471 (48:5813)
32.: R.K. Meyer, R.Z. Parks, Independent axioms for the implicational fragment of Sobocinski's three-valued logic, Zeitschr. f. math. Logik und Grundlagen der Math. 18 (1972), 291-295. MR 0357062 (50:9530)
33.: I. Nishimura, On formulas of one variable in intuitionistic propositional calculus, J. Symbolic Logic 25 (1960), 327-331. MR 0142456 (26:25)
34.: J.S. Olson, Subdirectly irreducible residuated semilattices and positive universal classes, Studia Logica, 83 (2006), 393-406. MR 2250117
35.: J.S. Olson, J.G. Raftery, Positive Sugihara monoids, Algebra Universalis, to appear.
36.: H. Ono, Proof-theoretic methods in nonclassical logic -- an introduction, in: M. Takahashi, M. Okada and M. Dezani-Ciancaglini (eds.), ``Theories of Types and Proofs'', MSJ Memoirs 2, Mathematical Society of Japan, 1998, pp. 207-254. MR 1728763 (2000k:03125)
37.: H. Ono, Substructural logics and residuated lattices - an introduction, in V.F. Hendricks, J. Malinowski (eds.), ``50 Years of Studia Logica'', Trends in Logic, vol. 20, Kluwer, 2003, pp.177-212. MR 2045284 (2004k:03046)
38.: L. Rieger, Zametka o t. naz. svobodnyh algebrah s zamykanijami, Czechoslovak Math. J. 7 (82) (1957), 16-20. MR 0091266 (19:938c)
39.: P. Schroeder-Heister, K. Dosen (eds.), ``Substructural Logics'', Clarendon Press, Oxford, 1993. MR 1283190 (95a:03003)
40.: T. Sugihara, Strict implication free from implicational paradoxes, Memoirs of the Faculty of Liberal Arts, Fukui University, Series I (1955), 55-59.
41.: A.S. Troelstra, ``Lectures on Linear Logic'', CSLI Lecture Notes No. 29, 1992. MR 1163373 (93i:03083)
42.: C.J. van Alten, The finite model property for knotted extensions of propositional linear logic, J. Symbolic Logic 70 (2005), 84-98. MR 2119124 (2005m:03127)
43.: C.J. van Alten, J.G. Raftery, Rule separation and embedding theorems for logics without weakening, Studia Logica 76 (2004), 241-274. MR 2072985 (2005e:03132)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03B47, 03G25, 06D99, 06F05, 08A50, 08C15

J. G. Raftery
Affiliation: School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4001, South Africa
Email: raftery@ukzn.ac.za

DOI: 10.1090/S0002-9947-07-04235-3
PII: S 0002-9947(07)04235-3
Keywords: Locally finite variety, residuation, residuated lattice, representable, idempotent, Sugihara monoid, relative Stone algebra, relevance logic, mingle.
Received by editor(s): September 25, 2005
Posted: March 20, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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