zbMATH — the first resource for mathematics

Commutative integral bounded residuated lattices with an added involution. (English) Zbl 1181.03061
A symmetric residuated lattice is an algebra \(A=(A,\vee ,\wedge ,*,\rightarrow ,\sim,1,0)\) such that \((A,\vee ,\wedge ,*,\rightarrow ,1,0)\) is a commutative integral bounded residuated lattice and the equations \({\sim\sim x}=x\) and \({\sim(x\vee y)}={\sim x}\wedge{\sim y}\) are satisfied.
The aim of this paper is to investigate the properties of the unary operation \(\varepsilon \) defined by \(\varepsilon x={\sim x} \rightarrow 0\).
The authors give necessary and sufficient conditions for \(\varepsilon\) to be an interior operator. Since these conditions are rather restrictive, they consider when an iteration of \(\varepsilon \) is an interior operator. In particular they consider the chain of varieties of symmetric residuated lattices such that the \(n\)-fold iteration of \(\varepsilon\) is a Boolean interior operator. They show that these varieties are semisimple. For \(n=1\), the variety of symmetric Stonean residuated lattices is obtained. Also, the authors characterize the subvarieties admitting representations as subdirect products of chains.

03G25 Other algebras related to logic
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B52 Fuzzy logic; logic of vagueness
06F05 Ordered semigroups and monoids
Full Text: DOI
[1] Baaz, M., Infinite-valued Gödel logics with 0-1 projections and relativizations, (), 23-33 · Zbl 0862.03015
[2] Balbes, R.; Dwinger, P., Distributive lattices, (1974), University of Missouri Press Columbia, Miss · Zbl 0321.06012
[3] Boixader, D., Some properties concerning the quasi-inverse of a \(t\)-norm, Mathware soft comput., 5, 5-12, (1998) · Zbl 0932.03065
[4] Burris, S.; Sankappanavar, H.P., A course in universal algebra, (1981), Springer-Verlag New York · Zbl 0478.08001
[5] Cignoli, R., Free algebras in varieties of Stonean residuated lattices, Soft comput., 12, 315-320, (2008) · Zbl 1142.06003
[6] Cignoli, R.; de Gallego, M.S., The lattice structure of some łukasiewicz algebras, Algebra universalis, 13, 315-328, (1981) · Zbl 0495.03045
[7] Cignoli, R.; Esteva, F.; Godo, L.; Torrens, A., Basic logic is the logic of continuous \(t\)-norms and their residua, Soft comput., 4, 106-112, (2000)
[8] Cignoli, R.; Esteva, F.; Godo, L.; Montagna, F., On a class of left-continuous \(t\)-norms, Fuzzy sets and systems, 131, 283-296, (2002) · Zbl 1012.03032
[9] Cignoli, R.; Torrens, A., Standard completeness of Hájek basic logic and decompositions of BL-chains, Soft comput., 9, 862-868, (2005) · Zbl 1094.03013
[10] Cintula, P.; Klement, E.P.; Mesiar, R.; Navara, M., Residuated logics based on strict triangular norms with an involutive negation, Math. log. Q., 52, 269-282, (2006) · Zbl 1165.03326
[11] Esteva, F.; Godo, L., Monoidal \(t\)-norm based logic: towards a logic for left-continuous \(t\)-norms, Fuzzy sets and systems, 124, 271-288, (2001) · Zbl 0994.03017
[12] Esteva, F.; Godo, L.; Hájek, P.; Navara, M., Residuated fuzzy logics with an involutive negation, Arch. math. logic, 39, 103-124, (2000) · Zbl 0965.03035
[13] Esteva, F.; Godo, L.; García-Cerdeña, A., On the hierarchy of \(t\)-norm based residuated fuzzy logics, (), 251-272 · Zbl 1038.03026
[14] Flaminio, T.; Marchioni, E., T-norm based logic with an independent involutive negation, Fuzzy sets and systems, 157, 3125-3144, (2006) · Zbl 1114.03015
[15] Freytes, H., Injectives in residuated algebras, Algebra universalis, 51, 373-393, (2004) · Zbl 1079.06014
[16] Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., ()
[17] Grätzer, G., General lattice theory, (1978), Birkhäuser Verlag Basel, Stuttgar · Zbl 0385.06015
[18] Hájek, P., Methamathematics of fuzzy logic, (1998), Kluwer Dordrecht, Boston, London
[19] Hájek, P., Basic fuzzy logic and BL-algebras, Soft comput., 2, 124-128, (1998)
[20] Höle, U., Commutative, residuated l-monoids, (), 53-106 · Zbl 0838.06012
[21] Jenei, S.; Montagna, F., A proof of standard completeness for esteva and godo’s logic MTL, Studia. logica., 70, 183-192, (2002) · Zbl 0997.03027
[22] T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at logics without contraction, Preliminary report
[23] Moisil, G., Logique modale, Disquisit. math. phys., 2, 3-98, (1942), Reproduced in [24, pp. 341-431] · Zbl 0063.04064
[24] Moisil, G., Essais sur LES logiques non chrysippiennes, (1972), Editions de l’Academie de la Republique Socialiste de Roumanie Bucharest · Zbl 0241.02006
[25] Monteiro, A.A., Sur LES algèbres de Heyting symmétriques, Port. math., 39, 1-237, (1980)
[26] Sankappanavar, H.P., Heyting algebras with a dual lattice endomorphism, Z. math. logik grundlag. math., 33, 565-573, (1987) · Zbl 0633.06005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.