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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9947-04-03654-2
PII: S 0002-9947(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