# Article

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Keywords:
categorical equivalence; bounded \BCK-algebra; \MV-algebra; \DRl-semigroup
Summary:
In the paper it is proved that the category of \MV-algebras is equivalent to the category of bounded \DRl-semigroups satisfying the identity \$1-(1-x)=x\$. Consequently, by a result of D. Mundici, both categories are equivalent to the category of bounded commutative \BCK-algebras.
References:
[1] C. C. Chang: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490. DOI 10.1090/S0002-9947-1958-0094302-9 | MR 0094302 | Zbl 0084.00704
[2] C. C. Chang: A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 93 (1959), 74-80. MR 0122718 | Zbl 0093.01104
[3] R. Cignoli: Free lattice-ordered abelian groups and varieties of MV-algebras. Proc. IX. Latin. Amer. Symp. Math. Logic, Part 1, Not. Log. Mat. 38 (1993), 113-118. MR 1332526 | Zbl 0827.06012
[4] K. Iséki, S. Tanaka: An introduction to the theory of BCK-algebras. Math. Japonica. 23 (1978), 1-26. MR 0500283
[5] T. Kovář: A general theory of dually residuated lattice ordered monoids. Thesis, Palacky Univ. Olomouc, 1996.
[6] T. Kovář: Two remarks on dually residuated lattice ordered semigroups. Math. Slovaca. To appear. MR 1804468
[7] F. Lacava: Some properties of L-algebras and existencially closed L-algebras. Boll. Un. Mat. Ital., A(5) 16 (1979), 360-366. (In Italian.) MR 0541775
[8] D. Mundici: Interpretation of AF C*-algebras in Lukasiewicz sentential calculus. J. Funct. Analys. 65 (1986), 15-63. DOI 10.1016/0022-1236(86)90015-7 | MR 0819173
[9] D. Mundici: MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Math. Japonica 31 (1986), 889-894. MR 0870978 | Zbl 0633.03066
[10] J. Rachůnek: DRI-semigroups and MV-algebras. Czechoslovak Math. J. 123 (1998), 365-372. DOI 10.1023/A:1022801907138 | MR 1624268
[11] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105-114. DOI 10.1007/BF01360284 | MR 0183797 | Zbl 0138.02104
[12] S. Tanaka: On \$\Lambda\$-commutative algebras. Math. Sem. Notes Kobe 3 (1975), 59-64. MR 0419222 | Zbl 0324.02053
[13] T. Traczyk: On the variety of bounded commutative BCK-algebras. Math. Japonica 24 (1979), 283-292. MR 0550212 | Zbl 0422.03038
[14] H. Yutani: Quasi-commutative BCK-algebras and congruence relations. Math. Sem. Notes Kobe 5 (1977), 469-480. MR 0498112 | Zbl 0375.02053

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