Previous |  Up |  Next

Article

Keywords:
pseudo $BL$-algebra; $DR\ell $-monoid; filter; polar; representable pseudo $BL$-algebra
Summary:
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.
References:
[1] A. Dvurečenskij: On pseudo $MV$-algebras. Soft Computing 5 (2001), 347–354. DOI 10.1007/s005000100136 | Zbl 0998.06010
[2] A. Dvurečenskij: States on pseudo $MV$-algebras. Studia Logica 68 (2001), 301–327. DOI 10.1023/A:1012490620450 | MR 1865858 | Zbl 0999.06011
[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 1817439
[5] G. Grätzer: General Lattice Theory. Birkhäuser, Berlin, 1998. MR 1670580
[6] P. Hájek: Basic fuzzy logic and $BL$-algebras. Soft Computing 2 (1998), 124–128. DOI 10.1007/s005000050043
[7] P. Hájek: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam, 1998. MR 1900263
[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 10.1023/A:1021766309509 | Zbl 1012.06012
[11] J. Rachůnek: A duality between algebras of basic logic and bounded representable $DR\ell $-monoids. Math. Bohem. 126 (2001), 561–569. MR 1970259
[12] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. DOI 10.1007/BF01360284 | MR 0183797 | Zbl 0138.02104
Partner of
EuDML logo