A lattice-theoretical approach to extensions of filters in algebras of substructural logic

Document Type : Original Article


1 Department of Mathematical Methods in economy, Faculty of Economics, VSB-Technical University Ostrava, Sokolska 33, 701 21 Ostrava, Czech Republic

2 Department of Algebra and Geometry, Faculty of Sciences, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic


Commutative bounded integral residuated lattices (residutaed lattices, in short) form a large class of algebras containing algebras which are algebraic counterparts of certain propositional fuzzy logics. The paper deals with the so-called extended filters of filters of residuated lattices. It is used the fact that the extended filters of filters associated with subsets coincide with those associated ones with corresponding filters. This makes it possible to investigate the set of all extended filters of residuated lattices within the Heyting algebras of their filters by means of the structural methods of the theory of such algebras.


[1] R. Balbes, P. Dwinger, Distributive lattices, University of Missouri Press, Columbia, Missouri,
[2] R.L.O. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic foundation of many-valued reasoning,
Kluwer Academic Publishers, Dordrecht – Boston – London, 2000.
[3] L.C. Ciungu, Classes of residuated lattices, Annals of the University of Craiova, Mathematics
and Computer Science Series, 33 (2006), 189–207.
[4] A. Dvurečenskij, J. Rachůnek, Probabilistic averaging in bounded commutative residuated ℓ-
monoids, Discrete Mathematics, 306 (2006), 1317–1326.
[5] F. Esteva, L. Godo, Monoidal t-norm based logic: Towards a logic for left-continuous t-norms,
Fuzzy Sets and Systems, 124 (2001), 271–288.
[6] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at
substructural logics, Elsevier Studies in Logic and Foundations. Elsevier, Amsterdam, 2007.
[7] P. Hájek, Metamathematics of fuzzy logic, Kluwer, Amsterdam, 1998.
[8] M. Haveshki, M. Mohamadhasani, Extended filters in bounded commutative Rℓ-monoids, Soft
Computing, 16 (2012), 2165–2173.
[9] P. Jipsen, C. Tsinakis, A survey of residuated lattices, In: Ordered Algebraic Structures (ed.
J. Martinez), Kluwer Acad. Publ. Dordrecht, 2002, 19–56.
[10] T. Katriňák, Remarks on the W.C. Nemitz’s paper ”Semi-Boolean lattices”, Notre Dame
Journal of Formal Logic, 11 (1970), 425–430.