On the free S^{ω}_{1} -algebras

Document Type : Original Article

Authors

1 Department of Mathematics, University of Salerno, Italy

2 Department of Mathematics, Tbilisi State University, Georgia

3 Institute of Cybernetics, Georgian Technical University, Georgia

Abstract

One and two-generated free MV -algebras are algebraically described in the variety generated by perfect
MV -algebras.

Keywords


[1] S. Aguzzoli, S. Bova, V. Marra, Applications of finite duality to locally finite varieties of BLalgebras, Proceedings of the Symposium on Logical Foundations of Computer Science, LFCS
2009, Deerfield Beach, Florida, USA. Lecture Notes in Computer Science, 5407 Springer,
(2009), 1-15.
[2] P.L. Belluce, C.C. Chang, A weak completeness theorem for infinite valued predicate logic,
Journal of Symbolic Logic, 28 (1963), 43–50.
[3] P.L. Belluce, A. Di Nola, The MV -algebra of first order Lukasiewicz logic, Tatra Mountains
Mathematical Publications, 27 (2003), 7–22.
[4] S. Burris, H.P. Sankappanavar, A course in universal algebras, The Millenium Eddition, 2000.
[5] C.C. Chang, Algebraic analysis of many-valued logics, Transactions of the American Mathematical Society, 88 (1958), 467–490.
[6] C.C. Chang. A new proof of the completeness of the Lukasiewicz axioms, Transactions of the
American Mathematical Society, 93 (1959), 74–80.
[7] R.L.O. Cignoli, I.M.L. DOttaviano, D. Mundici, ´ Algebraic foundations of many-valued reasoning, volume 7 of Trends in Logic—Studia Logica Library. Kluwer Academic Publishers,
Dordrecht, 2000.
[8] A. Di Nola, R. Grigolia, Finiteness and duality in MV-algebras theory, Advances in soft computing, Lectures on Soft Computing and Fuzzy Logic, Physica-Verlag, A Springer-Verlag Company,
(2001), 71–88.
[9] A. Di Nola, R. Grigolia, MV-algebras in duality with labeled root systems, Discrete Mathematics,
243 (2002), 79–90.
[10] A. Di Nola, R. Grigolia, Profinite MV-spaces, Discrete Mathematics, 283(1-3) (2004), 61–69.
[11] A. Di Nola, R. Grigolia, G¨odel spaces and perfect MV-algebras, Journal of Applied Logic,
13(3) (2015), 270–284.
[12] A. Di Nola, R. Grigolia, G. Panti, Finitely generated free MV-algebras and their automorphism
groups, Studia Logica, 61(1) (1998), 65–78.
[13] A. Di Nola, G. Lenzi, G. Vitale, The geometry of free MV-algebras in Chang variety: A bridge
from semisimplicity to infinitesimals, (Submitted).
[14] A. Di Nola, G. Lenzi, G. Vitale, Geometry of super-Lukasiewicz logic, (Submitted).
[15] A. Di Nola, A. Lettieri, Perfect MV-algebras are categorically equivalent to Abelian `-groups,
Studia Logica, 88 (1994), 467–490.
[16] A. Di Nola, A. Lettieri, Equational characterization of all varieties of MV-algebras, Journal
of Algebra, 221 (1999), 463–474.
[17] R. Grigolia, Algebraic analysis of Lukasiewicz-Tarskis n-valued logical systems, In R. W´ojcicki
and G. Malinowski, editors, Selected papers on Lukasiewicz sentential calculus, pages 81-92.
Polish Academy of Sciences, Ossolineum, Wroclaw, 1977.
[18] R. Grigolia, Free algebras of non-classical logics, Monograph, ”Metsniereba”,
Tbilisi, (1987), 110 pp. (Russian)
[19] Y. Komori. Super- Lukasiewicz propositional logic, Nagoya Mathematical Journal, 84 (1981),
119–133.
[20] A.I. Mal´cev, Algebraic systems, Springer-Verlag Berlin Heidelberg, New York, 1973.
[21] R. McNaughton, A theorem about infinite-valued sentential logics, The Journal of Symbolic
Logic, 16(1) (1951), 1–13.
[22] D. Mundici, Interpretation of AF C

-algebras in Lukasiewicz sentential calculus, Journal of
Functional Analysis, 65 (1986), 15–63.
[23] G. Panti, Varieties of MV-algebras, Journal of Applied Non-Classical Logics, 9(1) (1999),
141–157.
[24] H.A. Priestley, Representation of distributive lattices by means of ordered stone spaces, Bulletin of the London Mathematical Society, 2 (1972), 186–190.
[25] B. Scarpellini, Die nichaxiomatisierbarkeit des unendlichwertigen pradikatenkalkulus von
 Lukasiewicz, Journal of Symbolic Logic, 27 (1962), 159–170.