Lexicographic extensions of dually residuated lattice ordered monoids

Jiří Rachůnek; Dana Šalounová

Mathematica Bohemica (2004)

  • Volume: 129, Issue: 3, page 283-295
  • ISSN: 0862-7959

Abstract

top
Dually residuated lattice ordered monoids ( D R -monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings ( M V -algebras, B L -algebras) and their non-commutative variants ( G M V -algebras, pseudo B L -algebras). In the paper, lex-extensions and lex-ideals of D R -monoids (which need not be commutative or bounded) satisfying a certain natural condition are studied.

How to cite

top

Rachůnek, Jiří, and Šalounová, Dana. "Lexicographic extensions of dually residuated lattice ordered monoids." Mathematica Bohemica 129.3 (2004): 283-295. <http://eudml.org/doc/249409>.

@article{Rachůnek2004,
abstract = {Dually residuated lattice ordered monoids ($DR\ell $-monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings ($MV$-algebras, $BL$-algebras) and their non-commutative variants ($GMV$-algebras, pseudo $BL$-algebras). In the paper, lex-extensions and lex-ideals of $DR\ell $-monoids (which need not be commutative or bounded) satisfying a certain natural condition are studied.},
author = {Rachůnek, Jiří, Šalounová, Dana},
journal = {Mathematica Bohemica},
keywords = {$DR\ell $-monoid; ideal; lex-extension; lex-ideal; algebras of fuzzy logics; monoid; ideal; lex-extension; lex-ideal; algebras of fuzzy logics},
language = {eng},
number = {3},
pages = {283-295},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Lexicographic extensions of dually residuated lattice ordered monoids},
url = {http://eudml.org/doc/249409},
volume = {129},
year = {2004},
}

TY - JOUR
AU - Rachůnek, Jiří
AU - Šalounová, Dana
TI - Lexicographic extensions of dually residuated lattice ordered monoids
JO - Mathematica Bohemica
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 129
IS - 3
SP - 283
EP - 295
AB - Dually residuated lattice ordered monoids ($DR\ell $-monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings ($MV$-algebras, $BL$-algebras) and their non-commutative variants ($GMV$-algebras, pseudo $BL$-algebras). In the paper, lex-extensions and lex-ideals of $DR\ell $-monoids (which need not be commutative or bounded) satisfying a certain natural condition are studied.
LA - eng
KW - $DR\ell $-monoid; ideal; lex-extension; lex-ideal; algebras of fuzzy logics; monoid; ideal; lex-extension; lex-ideal; algebras of fuzzy logics
UR - http://eudml.org/doc/249409
ER -

References

top
  1. Groupes et Anneaux Réticulés, Springer, Berlin, 1977. (1977) MR0552653
  2. Foundations of Many-Valued Reasoning, Kluwer Acad. Publ., Dordrecht, 2000. (2000) MR1786097
  3. Lex-subgroups of lattice ordered groups, Czechoslovak Math. J. 18 (1968), 86–103. (1968) Zbl0155.05902MR0225697
  4. Pseudo B L -algebras: Part I, Multiple-Valued Logic 8 (2002), 673–714. (2002) MR1948853
  5. Partially Ordered Groups, World Scientific, Singapore, 1999. (1999) Zbl0933.06010MR1791008
  6. Pseudo M V -algebras, Multiple Valued Logic 6 (2001), 95–135. (2001) MR1817439
  7. Metamathematics of Fuzzy Logic, Kluwer, Amsterdam, 1998. (1998) MR1900263
  8. Lex ideals of generalized M V -algebras, C. S. Calude, M. J. Dinneen, S. Sburlan (eds.), Combinatorics, Computability and Logic, Proc. DMTCS’01, Springer, London, 2001, pp. 125–136. (2001) MR1934826
  9. A general theory of dually residuated lattice ordered monoids, Thesis, Palacký Univ. Olomouc, 1996. (1996) 
  10. Pseudo B L -algebras and D R -monoids, Math. Bohem. 128 (2003), 199–208. (2003) Zbl1024.06005MR1995573
  11. Ideals of noncommutative D R -monoids, (to appear). (to appear) MR2121658
  12. Prime ideals and polars in D R -monoids and pseudo B L -algebras, Math. Slovaca 53 (2003), 233–246. (2003) MR2025020
  13. Representable dually residuated lattice ordered monoids, (to appear). (to appear) MR2070377
  14. 10.1023/A:1022801907138, Czechoslovak Math. J. 48 (1998), 365–372. (1998) DOI10.1023/A:1022801907138
  15. M V -algebras are categorically equivalent to a class of D R -semigroups, Math. Bohem. 123 (1998), 437–441. (1998) MR1667115
  16. A duality between algebras of basic logic and bounded representable D R -monoids, Math. Bohem. 126 (2001), 561–569. (2001) MR1970259
  17. 10.1023/A:1021766309509, Czechoslovak Math. J. 52 (2002), 255–273. (2002) Zbl1012.06012MR1905434DOI10.1023/A:1021766309509
  18. Direct decompositions of dually residuated lattice ordered monoids, (to appear). (to appear) MR2118156
  19. Lex-ideals of D R -monoids and algebras, Math. Slovaca 53 (2003), 321–330. (2003) MR2025465
  20. 10.1007/BF01360284, Math. Ann. 159 (1965), 105–114. (1965) Zbl0138.02104MR0183797DOI10.1007/BF01360284

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.