Abstract
In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬x ∨ ¬¬x = 1. Some applications are given.
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References
Balbes R., Dwinger P.: Distributive Lattices. University of Missouri Press, Columbia, Missouri (1974)
Blok W. J., Pigozzi D.: ‘On the structure of of varieties with equationally definable principal congruences I’. Algebra Universalis 15, 195–227 (1982)
Burris S., Sankappanavar H.P.: A Course in Universal Algebra, Graduate texts in Mathematics 78. Springer Verlag, New York (1981)
Busaniche M.: ‘Free algebras in varieties of BL-algebras generated by a chain’. Algebra universalis 50, 259–277 (2003)
Cignoli R.: ‘Free algebras in varieties of Stonean residuated lattices’. Soft Comput. 12, 315–320 (2008)
Cignoli R., Torrens A.: ‘Free Stone algebras’. Discrete Math 222, 251–257 (2002)
Cignoli R., Torrens A.: ‘Free algebras in varieties of BL-algebras with a Boolean retract’. Algebra Univers. 48, 55–79 (2002)
Cignoli R., Torrens A.: ‘Glivenko like theorems in natural expansions of BCK-logics’. Math. Log. Quart. 50, 2–111125 (2004)
Cignoli R., Torrens A.: ‘Free Algebras in Varieties of Glivenko MTL-algebras Satisfying the Equation 2(x 2) = (2x)2’. Studia Logica 83, 157–181 (2006)
Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices An Algebraic Glimpse at Subestructural Logics, Studies in Logic vol. 151. Elsevier, Amsterdam (2007)
Gispert J., Torrens A.: ‘Bounded BCK-algebras and their generated variety’. Math. Log. Quart., 53, 206–213 (2007)
Höhle, U., ‘Commutative, residuated l-monoids’, in U. Höhle and E.P. Klement (eds.), Non-Classical Logics and their Applications to Fuzzy Subsets: A Handbook on the Mathematical Foundations of Fuzzy Set Theory, Kluwer, Boston, 1995, pp. 53–106.
Kowalski, T., and H. Ono, Residuated Lattices: An algebraic glimpse at logics without contraction, Japan Advanced Institut of Science and Thechnology, March 2001.
Monteiro A.: ‘Sur les algèbres de Heyting symmétriques’. Port. Math 39, 1–237 (1980)
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Castaño, D., Díaz Varela, J.P. & Torrens, A. Free-decomposability in Varieties of Pseudocomplemented Residuated Lattices. Stud Logica 98, 223–235 (2011). https://doi.org/10.1007/s11225-011-9326-2
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DOI: https://doi.org/10.1007/s11225-011-9326-2