Representable dually residuated lattice-ordered monoids

Jan Kühr

Discussiones Mathematicae - General Algebra and Applications (2003)

  • Volume: 23, Issue: 2, page 115-123
  • ISSN: 1509-9415

Abstract

top
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.

How to cite

top

Jan Kühr. "Representable dually residuated lattice-ordered monoids." Discussiones Mathematicae - General Algebra and Applications 23.2 (2003): 115-123. <http://eudml.org/doc/287614>.

@article{JanKühr2003,
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.},
author = {Jan Kühr},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {DRl-monoid; ideal; prime ideal; polar; normal ideal; representable DRl-monoid; variety},
language = {eng},
number = {2},
pages = {115-123},
title = {Representable dually residuated lattice-ordered monoids},
url = {http://eudml.org/doc/287614},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Jan Kühr
TI - Representable dually residuated lattice-ordered monoids
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2003
VL - 23
IS - 2
SP - 115
EP - 123
AB - 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.
LA - eng
KW - DRl-monoid; ideal; prime ideal; polar; normal ideal; representable DRl-monoid; variety
UR - http://eudml.org/doc/287614
ER -

References

top
  1. [1] S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York 1981. 
  2. [2] A. Di Nola, G. Georgescu, and A. Iorgulescu, Pseudo BL-algebras: Part I, Mult.-Valued Logic 8 (2002), 673-714. Zbl1028.06007
  3. [3] A. Di Nola, G. Georgescu, and A. Iorgulescu, Pseudo BL-algebras: Part II, Mult.-Valued Logic 8 (2002), 717-750. Zbl1028.06008
  4. [4] A. Dvurecenskij, On pseudo MV-algebras, Soft Comput. 5 (2001), 347-354. Zbl0998.06010
  5. [5] A. Dvurecenskij, States on pseudo MV-algebras, Studia Logica 68 (2001), 301-327. Zbl0999.06011
  6. [6] G. Georgescu, and A. Iorgulescu, Pseudo MV- algebras, Mult.-Valued Logic 6 (2001), 95-135. Zbl1014.06008
  7. [7] A.M.W. Glass, Partially Ordered Groups, World Scientific, Singapore-New Jersey-London-Hong Kong 1999. Zbl0933.06010
  8. [8] G. Grätzer, General Lattice Theory, Birkhäuser, Basel-Boston-Berlin 1998. Zbl0909.06002
  9. [9] M.E. Hansen, Minimal prime ideals in autometrized algebras, Czechoslovak Math. J. 44 (119) (1994), 81-90. Zbl0814.06011
  10. [10] T. Kovár, A general theory of dually residuated lattice-ordered monoids, Ph.D. thesis, Palacký University, Olomouc 1996. 
  11. [11] T. Kovár, Two remarks on dually residuated lattice-ordered semigroups, Math. Slovaca 49 (1999), 17-18. Zbl0943.06007
  12. [12] J. Kühr, Ideals of non-commutative DRl-monoids, Czechoslovak Math. J., to appear. 
  13. [13] J. Kühr, Pseudo BL-algebras and DRl-monoids, Math. Bohem. 128 (2003), 199-208. Zbl1024.06005
  14. [14] J. Kühr, Prime ideals and polars in DRl-monoids and pseudo BL-algebras, Math. Slovaca 53 (2003), 233-246. Zbl1058.06017
  15. [15] J. Rach unek, MV-algebras are categorically equivalent to a class of DRl1(i)-semigroups, Math. Bohem. 123 (1998), 437-441. 
  16. [16] J. Rach unek, A duality between algebras of basic logic and bounded representable DRl-monoids, Math. Bohem. 126 (2001), 561-569. 
  17. [17] J. Rach unek, A non-commutative generalization of MV-algebras, Czechoslovak Math. J. 52 (127) (2002), 255-273. 
  18. [18] K.L.N. Swamy, Dually residuated lattice-ordered semigroups. I, Math. Ann. 159 (1965), 105-114. Zbl0135.04203
  19. [19] K.L.N. Swamy, Dually residuated lattice-ordered semigroups. III, Math. Ann. 167 (1966), 71-74. Zbl0158.02601

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.