Previous |  Up |  Next

Article

Keywords:
$BL$-algebra; $MV$-algebra; bounded $DRl$-monoid; representable $DRl$-monoid; prime spectrum
Summary:
$BL$-algebras, introduced by P. Hájek, form an algebraic counterpart of the basic fuzzy logic. In the paper it is shown that $BL$-algebras are the duals of bounded representable $DRl$-monoids. This duality enables us to describe some structure properties of $BL$-algebras.
References:
[1] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. Springer, Berlin, 1977. MR 0648287
[2] Chang, C. C.: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88 (1958), 456–490. MR 0094302 | Zbl 0084.00704
[3] Chang, C. C.: A new proof of the completeness of the Łukasiewicz axioms. Trans. Amer. Math. Soc. 93 (1959), 74–80. MR 0122718 | Zbl 0093.01104
[4] Cignoli, R.: Lattice-ordered abelian groups and completeness of Łukasiewicz and product $t$-norm. Sémin. Structures Alg. Ord., Univ. Paris VII, No. 68, 1999, pp. 14.
[5] Cignoli, R., Torrens, A.: The poset of prime $l$-ideals of an abelian $l$-group with a strong unit. J. Algebra 184 (1996), 604–614. MR 1409232
[6] Hájek, P.: Basic fuzzy logic and $BL$-algebras. Inst. Comput. Sciences Acad. Sci. Czech Rep., Techn. Report V736 (1997).
[7] Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam, 1998. MR 1900263
[8] Mundici, D.: Interpretation of AF $C^{*}$-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15–63. MR 0819173 | Zbl 0597.46059
[9] Mundici, D.: $MV$-algebras are categorically equivalent to bounded commutative $BCK$-algebras. Math. Japonica 31 (1986), 889–894. MR 0870978 | Zbl 0633.03066
[10] Kovář, T.: A general theory of dually residuated lattice ordered monoids. Thesis, Palacký Univ., 1996.
[11] Rachůnek, J.: Prime ideals in autometrized algebras. Czechoslovak Math. J. 37 (1987), 65–69. MR 0875128
[12] Rachůnek, J.: $DRl$-semigroups and $MV$-algebras. Czechoslovak Math. J. 48 (1998), 365–372.
[13] Rachůnek, J.: Spectra of autometrized lattice algebras. Math. Bohem. 123 (1998), 87–94. MR 1618727
[14] Rachůnek, J.: $MV$-algebras are categorically equivalent to a class of $DRl_{1(i)}$-semigroups. Math. Bohem. 123 (1998), 437–441. MR 1667115
[15] Rachůnek, J.: Ordered prime spectra of bounded $DRl$-monoids. Math. Bohem (to appear). MR 1802299
[16] Swamy, K. L. N.: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. MR 0183797 | Zbl 0138.02104
[17] Swamy, K. L. N.: Dually residuated lattice ordered semigroups II. Math. Ann. 160 (1965), 64–71. MR 0191851 | Zbl 0138.02104
[18] Swamy, K. L. N.: Dually residuated lattice ordered semigroups III. Math. Ann. 167 (1966), 71–74. MR 0200364 | Zbl 0158.02601
[19] Swamy, K. L. N., Rao, N. P.: Ideals in autometrized algebras. J. Austral. Math. Soc. 24 (1977), 362–374. MR 0469843
[20] Swamy, K. L. N., Subba Rao, B. V.: Isometries in dually residuated lattice ordered semigroups. Math. Sem. Notes 8 (1980), 369–380. MR 0601906
[21] Turunen, E.: Boolean deductive systems of $BL$-algebras. Research Report 61, Lappeenranta Univ. of Technology, Dept. Inf. Techn., 1998.
Partner of
EuDML logo