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On the finite embeddability property for residuated ordered groupoids


Authors: W. J. Blok and C. J. van Alten
Journal: Trans. Amer. Math. Soc. 357 (2005), 4141-4157
MSC (2000): Primary 06F05; Secondary 03B47, 06F07, 06F99, 08C15, 08A50
DOI: https://doi.org/10.1090/S0002-9947-04-03654-2
Published electronically: October 7, 2004
MathSciNet review: 2159703
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Abstract: The finite embeddability property (FEP) for integral, commutative residuated ordered monoids was established by W. J. Blok and C. J. van Alten in 2002. Using Higman's finite basis theorem for divisibility orders we prove that the assumptions of commutativity and associativity are not required: the classes of integral residuated ordered monoids and integral residuated ordered groupoids have the FEP as well. The same holds for their respective subclasses of (bounded) (semi-)lattice ordered structures. The assumption of integrality cannot be dropped in general--the class of commutative, residuated, lattice ordered monoids does not have the FEP--but the class of $n$-potent commutative residuated lattice ordered monoids does have the FEP, for any $n < \omega$.


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  • 1. G. Birkhoff, ``Lattice Theory", Amer. Math. Soc., Providence, 1967.MR 82a:06001
  • 2. W. J. Blok, D. Pigozzi, ``Algebraizable logics", Memoirs of the American Mathematical Society, 396, Amer. Math. Soc., Providence, 1989. MR 90d:03140
  • 3. W.J. Blok, J.G. Raftery, Varieties of commutative residuated integral pomonoids and their residuation subreducts, J. Algebra 190 (1997), 280-328. MR 98j:08001
  • 4. 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 2003j:06017
  • 5. T.S. Blyth, M.F. Janowitz, ``Residuation Theory", Pergamon Press, Oxford-New York 1972. MR 53:226
  • 6. Buszkowski, W., Mathematical linguistics and proof theory, in ``Handbook of Logic and Language", J. van Benthem and A. ter Meulen, editors, North-Holland Publishing Co., Amsterdam; MIT Press, Cambridge, MA, 1997. MR 98e:03025
  • 7. Dilworth, R. P, Non-commutative residuated lattices, Trans. Amer. Math. Soc. 46, (1939), 426-444. MR 1:37d
  • 8. T. Evans, Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc. 1 (1969), 399-403. MR 40:2589
  • 9. I.M.A. Ferreirim, On varieties and quasivarieties of hoops and their reducts, Ph.D. thesis, University of Illinois at Chicago, 1992.
  • 10. L. Fuchs, Partially Ordered Algebraic Systems, Addison-Wesley, 1963. MR 30:2090
  • 11. J. A. Green, D. Rees, On semi-groups in which $x\sp r=x$, Proc. Cambridge Philos. Soc. 48 (1952), 35-40. MR 13:720c
  • 12. G. Higman, Ordering by divisibility in abstract algebras, Proc. London Math. Soc. (3) 2 (1952) 326-336. MR 14:238e
  • 13. D. Higgs, Dually residuated commutative monoids with identity element do not form an equational class, Math. Japon. 29 (1984), 69-75. MR 86a:06021
  • 14. Y. Lafont, The finite model property for various fragments of linear logic, J. Symbolic Logic 62 (1997), 1202-1208. MR 99m:03121
  • 15. J.C.C. McKinsey, A. Tarski, On closed elements in closure algebras, Ann. of Math. 47 (1946), 122-162. MR 7:359e
  • 16. M. Okada, K. Terui, The finite model property for various fragments of intuitionistic linear logic, J. Symbolic Logic 64 (1999) no. 2, 790-802. MR 2002g:03043
  • 17. H. Ono, Y. Komori, Logics without the contraction rule, J. Symbolic Logic 50 (1985), 169-202. MR 87a:03053
  • 18. C.J. van Alten, J.G. Raftery, On quasivariety semantics of fragments of intuitionistic propositional logic without exchange and contraction rules, Reports on Mathematical Logic 31 (1997), 3-55; also Internal Report No. 5/99, School of Mathematical and Statistical Sciences, University of Natal, Durban. MR 2001e:03023
  • 19. Rosenthal, Kimmo I. ``Quantales and their applications", Pitman Research Notes in Mathematics Series, 234. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990. MR 92e:06028
  • 20. M. Ward, R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939), 335-354.
  • 21. W. Wechler, ``Universal Algebra for Computer Scientists", Springer-Verlag, Berlin-Heidelberg, 1992. MR 94a:08001

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Additional Information

W. J. Blok
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045
Email: wjb@uic.edu

C. J. van Alten
Affiliation: School of Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa
Email: cvalten@maths.wits.ac.za

DOI: https://doi.org/10.1090/S0002-9947-04-03654-2
Keywords: Finite embeddability property, residuated ordered groupoid, residuated ordered monoid, residuated lattice, quantale, integrality, finite basis property, divisibility order, well-quasi-order
Received by editor(s): December 2, 2003
Published electronically: October 7, 2004
Additional Notes: Professor Willem Blok tragically passed away shortly after submitting this paper
Article copyright: © Copyright 2004 American Mathematical Society

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