Geodesic
Pseudo $BL$-algebras and $DR\ell $-monoids
Mathematica Bohemica, Tome 128 (2003) no. 2, pp. 199-208

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
It is shown that pseudo $BL$-algebras are categorically equivalent to certain bounded $DR\ell $-monoids. Using this result, we obtain some properties of pseudo $BL$-algebras, in particular, we can characterize congruence kernels by means of normal filters. Further, we deal with representable pseudo $BL$-algebras and, in conclusion, we prove that they form a variety.
It is shown that pseudo $BL$-algebras are categorically equivalent to certain bounded $DR\ell $-monoids. Using this result, we obtain some properties of pseudo $BL$-algebras, in particular, we can characterize congruence kernels by means of normal filters. Further, we deal with representable pseudo $BL$-algebras and, in conclusion, we prove that they form a variety.
DOI : 10.21136/MB.2003.134040
Classification : 03B52, 03G25, 06D35, 06F05
Keywords: pseudo $BL$-algebra; $DR\ell $-monoid; filter; polar; representable pseudo $BL$-algebra 
Kühr, Jan. Pseudo $BL$-algebras and $DR\ell $-monoids. Mathematica Bohemica, Tome 128 (2003) no. 2, pp. 199-208. doi: 10.21136/MB.2003.134040
@article{10_21136_MB_2003_134040,
     author = {K\"uhr, Jan},
     title = {Pseudo $BL$-algebras and $DR\ell $-monoids},
     journal = {Mathematica Bohemica},
     pages = {199--208},
     year = {2003},
     volume = {128},
     number = {2},
     doi = {10.21136/MB.2003.134040},
     mrnumber = {1995573},
     zbl = {1024.06005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2003.134040/}
}
TY  - JOUR
AU  - Kühr, Jan
TI  - Pseudo $BL$-algebras and $DR\ell $-monoids
JO  - Mathematica Bohemica
PY  - 2003
SP  - 199
EP  - 208
VL  - 128
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2003.134040/
DO  - 10.21136/MB.2003.134040
LA  - en
ID  - 10_21136_MB_2003_134040
ER  - 
%0 Journal Article
%A Kühr, Jan
%T Pseudo $BL$-algebras and $DR\ell $-monoids
%J Mathematica Bohemica
%D 2003
%P 199-208
%V 128
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2003.134040/
%R 10.21136/MB.2003.134040
%G en
%F 10_21136_MB_2003_134040

[1] A. Dvurečenskij: On pseudo $MV$-algebras. Soft Computing 5 (2001), 347–354. | DOI | MR | Zbl

[2] A. Dvurečenskij: States on pseudo $MV$-algebras. Studia Logica 68 (2001), 301–327. | DOI | MR | Zbl

[3] A. Di Nola, G. Georgescu, A. Iorgulescu: Pseudo $BL$-algebras: Part I. Preprint.

[4] G. Georgescu, A. Iorgulescu: Pseudo $MV$-algebras. Mult. Val. Logic 6 (2001), 95–135. | MR

[5] G. Grätzer: General Lattice Theory. Birkhäuser, Berlin, 1998. | MR

[6] P. Hájek: Basic fuzzy logic and $BL$-algebras. Soft Computing 2 (1998), 124–128. | DOI

[7] P. Hájek: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam, 1998. | MR

[8] T. Kovář: A general theory of dually residuated lattice ordered monoids. Ph.D. thesis, Palacký Univ., Olomouc, 1996.

[9] J. Kühr: Ideals of noncommutative $DR\ell $-monoids. Manuscript.

[10] J. Rachůnek: A non-commutative generalization of $MV$-algebras. Czechoslovak Math. J. 52 (2002), 255–273. | DOI | MR | Zbl

[11] J. Rachůnek: A duality between algebras of basic logic and bounded representable $DR\ell $-monoids. Math. Bohem. 126 (2001), 561–569. | MR

[12] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. | DOI | MR | Zbl

Cité par Sources :