# Lexicographic extensions of dually residuated lattice ordered monoids

Mathematica Bohemica (2004)

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

## Access Full Article

top## Abstract

top## How to cite

topRachůnek, 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- Groupes et Anneaux Réticulés, Springer, Berlin, 1977. (1977) MR0552653
- Foundations of Many-Valued Reasoning, Kluwer Acad. Publ., Dordrecht, 2000. (2000) MR1786097
- Lex-subgroups of lattice ordered groups, Czechoslovak Math. J. 18 (1968), 86–103. (1968) Zbl0155.05902MR0225697
- Pseudo $BL$-algebras: Part I, Multiple-Valued Logic 8 (2002), 673–714. (2002) MR1948853
- Partially Ordered Groups, World Scientific, Singapore, 1999. (1999) Zbl0933.06010MR1791008
- Pseudo $MV$-algebras, Multiple Valued Logic 6 (2001), 95–135. (2001) MR1817439
- Metamathematics of Fuzzy Logic, Kluwer, Amsterdam, 1998. (1998) MR1900263
- Lex ideals of generalized $MV$-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
- A general theory of dually residuated lattice ordered monoids, Thesis, Palacký Univ. Olomouc, 1996. (1996)
- Pseudo $BL$-algebras and $DR\ell $-monoids, Math. Bohem. 128 (2003), 199–208. (2003) Zbl1024.06005MR1995573
- Ideals of noncommutative $DR\ell $-monoids, (to appear). (to appear) MR2121658
- Prime ideals and polars in $DR\ell $-monoids and pseudo $BL$-algebras, Math. Slovaca 53 (2003), 233–246. (2003) MR2025020
- Representable dually residuated lattice ordered monoids, (to appear). (to appear) MR2070377
- $DR\ell $-semigroups and $MV$-algebras, Czechoslovak Math. J. 48 (1998), 365–372. (1998)
- $MV$-algebras are categorically equivalent to a class of $DR\ell $-semigroups, Math. Bohem. 123 (1998), 437–441. (1998) MR1667115
- A duality between algebras of basic logic and bounded representable $DR\ell $-monoids, Math. Bohem. 126 (2001), 561–569. (2001) MR1970259
- 10.1023/A:1021766309509, Czechoslovak Math. J. 52 (2002), 255–273. (2002) Zbl1012.06012DOI10.1023/A:1021766309509
- Direct decompositions of dually residuated lattice ordered monoids, (to appear). (to appear) MR2118156
- Lex-ideals of $DR\ell $-monoids and algebras, Math. Slovaca 53 (2003), 321–330. (2003) MR2025465
- 10.1007/BF01360284, Math. Ann. 159 (1965), 105–114. (1965) Zbl0138.02104MR0183797DOI10.1007/BF01360284

## NotesEmbed ?

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