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The Priestley duality for Wajsberg algebras. (English) Zbl 0717.03026
Wajsberg algebras are the algebraic counterpart of Łukasiewicz logic that is defined with axioms which characterize implication and negation and with Modus Ponens as a rule [A. J. Rodriguez, Un estudio algebraico de los cálculos proposicionales de Łukasiewicz. Ph. D. Thesis, Univ. Barcelona (1980)]. In this paper, Wajsberg algebras are considered as Kleene algebras and bounded distributive lattices. The aim of this paper is to develop a duality theory for Wajsberg algebras, extending the duality between Kleene algebras and some ordered topological spaces that is considered by W. Cornish and P. Fowler [J. Austral. Math. Soc., Ser. A 27, 209-220 (1978; Zbl 0403.06010)]. A Wajsberg space is characterized as Kleene space over a family of partial functions satisfying a number of algebraic and topological properties.
Reviewer: G.E.Tseytlin

03G25 Other algebras related to logic
03B50 Many-valued logic
06F30 Ordered topological structures
Full Text: DOI
[1] R. Cignoli, Proper n-valued ?ukasiewicz algebras as S-algebras of ?ukasiewicz n-valued propositional calculi, Studia Logica XLI (1982), pp. 3-16. · Zbl 0509.03012
[2] R. Cignoli, manuscript.
[3] W. Cornish and P. Fowler, Coproducts of De Morgan algebras, Bulletin of the Australian Mathematical Society 16 (1977), pp. 1-13. · Zbl 0329.06005
[4] W. Cornish and P. Fowler, Coproducts of Kleene algebras, J. Austral. Math. Soc. Ser A 27, pp. 209-220. · Zbl 0403.06010
[5] C. C. Chang, Algebraic Analysis of many valued logics, Transactions of the American Mathematical Society 88 (1958), pp. 467-490. · Zbl 0084.00704
[6] C. C. Chang, A new proof of the completeness of the ?ukasiewicz axioms, Transactions of the American Mathematical Society 93 (1959), pp. 74-80. · Zbl 0093.01104
[7] J. Font, A. Rodriguez and A. Torrens, Wajsberg algebras, Stochastica, vol. VIII, N? 1 (1984), pp. 5-31.
[8] D. Gluschankof and N. Martinez: The Kleene structure does not determinate the Wajsberg implication. Comunication to the U.M.A. (1987).
[9] D. Gluschankof: Doctoral Thesis (in preparation). Fac. Cs. Exactas y Naturales, Universidad de Buenos Aires.
[10] R. S. Grigolia: Algebraic analysis of ?ukasiewicz-Tarski n-valued logical systems, In Selected Papers on ?ukasiewicz Sentential Calculi, Ryszard W?jcicki ed., Warszawa, 1974.
[11] Y. Komori, The separation theorem of the X 0-valued ?ukasiewicz propositional logic, Rep. Fac. of Sc., Shizuoka University, vol. 12 (1978), pp. 1-5. · Zbl 0377.02021
[12] Y. Komori, Super ?ukasiewicz implicational logics, Nagoya Math. J., vol. 72 (1978), pp. 127-133. · Zbl 0363.02015
[13] Y. Komori, Super ?ukasiewicz implicational logics, Nagoya Math. J., vol. 84 (1981), pp. 119-133. · Zbl 0482.03007
[14] A. Monteiro: L’arithmetique des filtres et us espaces topologiques, Notas de L?gica mathem?tica, Univ. Nac. des Sur. 1974, pp. 29-30.
[15] D. Mundici: Interpretation of AF C *-algebras in ?ukasiewicz Sentential Calculus, Journal of Functional Analysis, 65, N? 1, January 1986. · Zbl 0597.46059
[16] H. A. Priestley: Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2, pp. 186-190. · Zbl 0201.01802
[17] H. A. Priestley: Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24, pp. 507-530. · Zbl 0323.06011
[18] A. J. Rodriguez: Un estudio algebraico de los c?lculas proposicionales de ?ukasiewicz, Ph. D. Thesis. Universidad de Barcelona, 1980.
[19] A. Torrens, W-algebras which are boolean products of members of SR [1], Studia Logica 46 (1987), pp. 265-275. · Zbl 0621.03042
[20] T. Traczyk, On the variety of bounded conmutative BCK-algebras, Math. Jap., 24 3 (1979), pp. 283-292. · Zbl 0422.03038
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