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 *-potent* commutative residuated lattice ordered monoids does have the FEP, for any .

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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