Abstract
We study Dedekind complete commutative BCK-algebras with the relative cancellation property and their connection with corresponding universal groups. We shall characterize Dedekind orthogonally complete atomic and Archimedean BCK-algebras, generalizing results of Jakubík known for MV-algebras. Finally, we characterize those Dedekind complete and atomic commutative BCK-algebras that are isomorphic to direct products of basic BCK-chains, generalizing a result of Cignoli for MV-algebras.
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References
Baer, R. (1949) Free sums of groups and their generalizations. An analysis of the associative law, Amer. J. Math. 41, 706-742.
Birkhoff, G. (1967) Lattice Theory, Amer. Math. Soc. Coll. Publ. 25, Providence, RI.
Cignoli, R. (1991) Complete and atomic algebras of the infinite valued Łukasiewicz logic, Studia Logica 50, 375-384.
Cornish, W. H., Sturm, T. and Traczyk, T. (1984) Embedding of commutative BCK-algebras into distributive lattice BCK-algebras, Math. Japonica 29, 309-320.
Dvurečenskij, A. (1998) Measures and states on BCK-algebras, Atti Sem. Mat. Fis. Univ. Modena 47, 267-284.
Dvurečenskij, A., Commutative BCK-algebras and lattice ordered groups with universal property, Indian J. Math., to appear.
Dvurečenskij, A. and Graziano, M. G. (1998) Remarks on representations of minimal clans, Tatra Mt. Math. Publ. 15, 31-53.
Dvurečenskij, A. and Graziano, M. G. (1999) On representations of commutative BCK-algebras, Demonstratio Math. 32, 227-246.
Dvurečenskij, A. and Graziano, M. G. (1999) Commutative BCK-algebras and lattice ordered groups, Math. Japonica 49, 159-174.
Goodearl, K. R. (1986) Partially Ordered Abelian Groups with Interpolation, Math. Surveys Monographs 20, Amer. Math. Soc., Providence, RI.
Imai, Y. and Iséki, K. (1966) On axiom systems of propositional calculi, Proc. Japan Acad. 42, 19-22.
Iséki, K. (1979) A special class of BCK-algebras, Math. Seminar Notes 5, 107-119.
Jakubík, J. (1975) Conditionally orthogonally complete ℓ-groups, Math. Nachr. 65, 153-162.
Jakubík, J. (1995) On complete MV-algebras, Czechoslovak Math. J. 45(120), 473-480.
Jakubík, J. (1998) On archimedean MV-algebras, Czechoslovak Math. J. 48, 75-582.
Jakubík, J., Completeness properties for MV-algebras (to appear).
Meng, J. and Jun, Y. B. (1994) BCK-algebras, Kyung Moon Sa Co., Seoul.
Mundici, D. (1986) MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japonica 31, 889-894.
Mundici, D. (1986) Interpretation of AF C*-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65, 15-63.
Palasiński, M. (1980) Some remarks on BCK-algebras, Math. Seminar Notes Univ. Kobe 8, 137-144.
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Dvurečenskij, A., Graziano, M.G. Dedekind Complete Commutative BCK-algebras. Order 17, 23–41 (2000). https://doi.org/10.1023/A:1006409215452
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DOI: https://doi.org/10.1023/A:1006409215452