PDF

Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 115-123
DOI: https://doi.org/10.7151/dmgaa.1067

REPRESENTABLE DUALLY RESIDUATED LATTICE-ORDERED MONOIDS

Jan Kühr

Department of Algebra and Geometry,
Faculty of Science, Palacký University
Tomkova 40, 779 00 Olomouc, Czech Republic
e-mail: kuhr@inf.upol.cz

Abstract

Dually residuated lattice-ordered monoids (DRl-monoids) generalize lattice-ordered groups and include also some algebras related to fuzzy logic (e.g. GMV-algebras and pseudo BL-algebras). In the paper, we give some necessary and sufficient conditions for a DRl-monoid to be representable (i.e. a subdirect product of totally ordered DRl-monoids) and we prove that the class of representable DRl-monoids is a variety.

Keywords: DRl-monoid, ideal, prime ideal, polar, normal ideal,representable DRl-monoid.

2000 Mathematics Subject Classification: 06F05, 03G25.

References

[1]	S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York 1981.
[2]	A. Di Nola, G. Georgescu, and A. Iorgulescu, Pseudo BL-algebras: Part I, Mult.-Valued Logic 8 (2002), 673-714.
[3]	A. Di Nola, G. Georgescu, and A. Iorgulescu, Pseudo BL-algebras: Part II, Mult.-Valued Logic 8 (2002), 717-750.
[4]	A. Dvurecenskij, On pseudo MV-algebras, Soft Comput. 5 (2001), 347-354.
[5]	A. Dvurecenskij, States on pseudo MV-algebras, Studia Logica 68 (2001), 301-327.
[6]	G. Georgescu, and A. Iorgulescu, Pseudo MV- algebras, Mult.-Valued Logic 6 (2001), 95-135.
[7]	A.M.W. Glass, Partially Ordered Groups, World Scientific, Singapore-New Jersey-London-Hong Kong 1999.
[8]	G. Grätzer, General Lattice Theory, Birkhäuser, Basel-Boston-Berlin 1998.
[9]	M.E. Hansen, Minimal prime ideals in autometrized algebras, Czechoslovak Math. J. 44 (119) (1994), 81-90.
[10]	T. Kovár, A general theory of dually residuated lattice-ordered monoids, Ph.D. thesis, Palacký University, Olomouc 1996.
[11]	T. Kovár, Two remarks on dually residuated lattice-ordered semigroups, Math. Slovaca 49 (1999), 17-18.
[12]	J. Kühr, Ideals of non-commutative DRl-monoids, Czechoslovak Math. J., to appear.
[13]	J. Kühr, Pseudo BL-algebras and DRl-monoids, Math. Bohem. 128 (2003), 199-208.
[14]	J. Kühr, Prime ideals and polars in DRl-monoids and pseudo BL-algebras, Math. Slovaca 53 (2003), 233-246.
[15]	J. Rach unek, MV-algebras are categorically equivalent to a class of DRl1(i)-semigroups, Math. Bohem. 123 (1998), 437-441.
[16]	J. Rach unek, A duality between algebras of basic logic and bounded representable DRl-monoids, Math. Bohem. 126 (2001), 561-569.
[17]	J. Rach unek, A non-commutative generalization of MV-algebras, Czechoslovak Math. J. 52 (127) (2002), 255-273.
[18]	K.L.N. Swamy, Dually residuated lattice-ordered semigroups. I, Math. Ann. 159 (1965), 105-114.
[19]	K.L.N. Swamy, Dually residuated lattice-ordered semigroups. III, Math. Ann. 167 (1966), 71-74.

Received 11 February 2003