| Title:
|
Negation in bounded commutative $DR\ell$-monoids (English) |
| Author:
|
Rachůnek, Jiří |
| Author:
|
Slezák, Vladimír |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
2 |
| Year:
|
2006 |
| Pages:
|
755-763 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The class of commutative dually residuated lattice ordered monoids ($DR\ell $-monoids) contains among others Abelian lattice ordered groups, algebras of Hájek’s Basic fuzzy logic and Brouwerian algebras. In the paper, a unary operation of negation in bounded $DR\ell $-monoids is introduced, its properties are studied and the sets of regular and dense elements of $DR\ell $-monoids are described. (English) |
| Keyword:
|
$DR\ell $-monoid |
| Keyword:
|
$MV$-algebra |
| Keyword:
|
$BL$-algebra |
| Keyword:
|
Brouwerian algebra |
| Keyword:
|
negation |
| MSC:
|
06D35 |
| MSC:
|
06F05 |
| idZBL:
|
Zbl 1164.06325 |
| idMR:
|
MR2291772 |
| . |
| Date available:
|
2009-09-24T11:37:43Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128102 |
| . |
| Reference:
|
[1] R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-valued Reasoning.Kluwer, Dordrecht, 2000. MR 1786097 |
| Reference:
|
[2] R. Cignoli and A. Torrens: Hájek basic fuzzy logic and Lukasiewicz infinite-valued logic.Arch. Math. Logic 42 (2003), 361–370. MR 2018087, 10.1007/s001530200144 |
| Reference:
|
[3] P. Hájek: Metamathematics of Fuzzy Logic.Kluwer, Amsterdam, 1998. MR 1900263 |
| Reference:
|
[4] J. Rachůnek: $DR\ell $-semigroups and $MV$-algebras.Czechoslovak Math. J. 48 (1998), 365–372. MR 1624268, 10.1023/A:1022801907138 |
| Reference:
|
[5] J. Rachůnek: $MV$-algebras are categorically equivalent to a class of $DR\ell _{1(i)}$-semigroups.Math. Bohem. 123 (1998), 437–441. MR 1667115 |
| Reference:
|
[6] J. Rachůnek: A duality between algebras of basic logic and bounded representable $DR\ell $-monoids.Math. Bohem. 126 (2001), 561–569. MR 1970259 |
| Reference:
|
[7] K. L. N. Swamy: Dually residuated lattice ordered semigroups.Math. Ann. 159 (1965), 105–114. Zbl 0138.02104, MR 0183797, 10.1007/BF01360284 |
| Reference:
|
[8] K. L. N. Swamy: Dually residuated lattice ordered semigroups II.Math. Ann. 160 (1965), 65–71. MR 0191851 |
| Reference:
|
[9] K. L. N. Swamy: Dually residuated lattice ordered semigroups III.Math. Ann. 167 (1966), 71–74. Zbl 0158.02601, MR 0200364, 10.1007/BF01361218 |
| Reference:
|
[10] K. N. Swamy and B. V. Subba Rao: Isometries in dually residuated lattice ordered semigroups.Math. Sem. Notes (Kobe) 8 (1980), 369–380. MR 0601906 |
| . |
