On the finite embeddability property for residuated ordered groupoids
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- by W. J. Blok and C. J. van Alten
- Trans. Amer. Math. Soc. 357 (2005), 4141-4157
- DOI: https://doi.org/10.1090/S0002-9947-04-03654-2
- Published electronically: October 7, 2004
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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 $n < \omega$.References
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 25, American Mathematical Society, Providence, R.I., 1979. MR 598630
- W. J. Blok and Don Pigozzi, Algebraizable logics, Mem. Amer. Math. Soc. 77 (1989), no. 396, vi+78. MR 973361, DOI 10.1090/memo/0396
- Willem J. Blok and James G. Raftery, Varieties of commutative residuated integral pomonoids and their residuation subreducts, J. Algebra 190 (1997), no. 2, 280–328. MR 1441951, DOI 10.1006/jabr.1996.6834
- W. J. Blok and C. J. van Alten, The finite embeddability property for residuated lattices, pocrims and BCK-algebras, Algebra Universalis 48 (2002), no. 3, 253–271. MR 1954775, DOI 10.1007/s000120200000
- T. S. Blyth and M. F. Janowitz, Residuation theory, International Series of Monographs in Pure and Applied Mathematics, Vol. 102, Pergamon Press, Oxford-New York-Toronto, 1972. MR 0396359
- Johan van Benthem and Alice ter Meulen (eds.), Handbook of logic and language, North-Holland Publishing Co., Amsterdam; MIT Press, Cambridge, MA, 1997. MR 1437147
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- Trevor Evans, Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc. (2) 1 (1969), 399–403. MR 249344, DOI 10.1112/jlms/s2-1.1.399
- I.M.A. Ferreirim, On varieties and quasivarieties of hoops and their reducts, Ph.D. thesis, University of Illinois at Chicago, 1992.
- L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Denis Higgs, Dually residuated commutative monoids with identity element as least element do not form an equational class, Math. Japon. 29 (1984), no. 1, 69–75. MR 737536
- Yves Lafont, The finite model property for various fragments of linear logic, J. Symbolic Logic 62 (1997), no. 4, 1202–1208. MR 1617973, DOI 10.2307/2275637
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- Mitsuhiro Okada and Kazushige Terui, The finite model property for various fragments of intuitionistic linear logic, J. Symbolic Logic 64 (1999), no. 2, 790–802. MR 1777787, DOI 10.2307/2586501
- Hiroakira Ono and Yuichi Komori, Logics without the contraction rule, J. Symbolic Logic 50 (1985), no. 1, 169–201. MR 780534, DOI 10.2307/2273798
- C. J. van Alten and J. G. Raftery, On quasivariety semantics of fragments of intuitionistic propositional logic without exchange and contraction rules, Rep. Math. Logic 31 (1997), 3–55 (1998). MR 1772641
- Kimmo I. Rosenthal, Quantales and their applications, Pitman Research Notes in Mathematics Series, vol. 234, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990. MR 1088258
- M. Ward, R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939), 335–354.
- Wolfgang Wechler, Universal algebra for computer scientists, EATCS Monographs on Theoretical Computer Science, vol. 25, Springer-Verlag, Berlin, 1992. MR 1177406, DOI 10.1007/978-3-642-76771-5
Bibliographic 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
- 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
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2159703