Abstract
Hájek introduced the logic \(BL_{vt}\) enriching the logic BL by a unary connective vt which is a formalization of Zadeh’s fuzzy truth value “very true”. \(\hbox{BL}_{vt}\) algebras, i.e., BL-algebras with unary operations, called vt-operators, which are among others subdiagonal, are an algebraic counterpart of \(BL_{vt}.\) Partially ordered commutative integral residuated monoids (pocrims) are common generalizations of both BL-algebras and Heyting algebras. The aim of our paper is to introduce and study algebraic properties of pocrims endowed by “very-true” and “very-false”-like operators.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bělohlávek R, Vychodil V (2005) Reducing the size of fuzzy concepts by hedges. In: The 2005 IEEE international conference on fuzzy systems, pp 663–668
Blok WJ, Raftery JG (1997) Varieties of commutative residuated integral pomonoids and their residuation subreducts. J Algebra 190:280–328
Chajda I, Vychodil V (2006) A note on residuated lattices with globalization. Int J Pure Appl Math 27(3):299–303
Chajda I, Halaš R (2008) Functional completeness of weak logics with a strict negation. Multiple Valued Logic Soft Comp 15(1):55–59
Chajda I, Eigenthaler G, Länger H (2003) Congruence classes in universal algebra. Heldermann Verlag, Lemgo
Di Nola A, Sessa S, Esteva F, Godo L, Garcia P (2002) The variety generated by perfect BL-algebras: an algebraic approach in a fuzzy logic setting. Ann Math Artif Intell 35:197–214
Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left continuous t-norms. Fuzzy Sets Syst 124:271–288
Ganter B, Wille R (1999) Formal concept analysis. Mathematical foundations. Springer, Berlin
Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht
Hájek P (2001) On very true. Fuzzy Sets Syst 124:329–333
Higgs D (1984) Dually residuated commutative monoids with identity element do not form an equational class. Math Jap 29:69–75
Höhle U (1995) Commutative, residuated l-monoids. In: Höhle U, Klement EP (eds) Non-classical logics and their applications to fuzzy subsets. Kluwer, Dordrecht, pp 53–106
Iorgulescu A (2004) Classes of BCK algebras-part I. In: Preprint series of the Institute of Mathematics of the Romanian Academy, preprint nr. 1/2004
Iséki K (1966) An algebra related to a propositional calculus. Proc Jpn Acad 42:26–29
Jipsen P, Tsinakis C (2002) A survey of residuated lattices. In: Martinez J (ed) Ordered algebraic structures. Kluwer, Dordrecht, pp 19–56
Kowalski T, Ono H (2001) Residuated lattices: an algebraic glimpse at logics without contraction, Monograph
Pałasiński M (1982) An embedding theorem for BCK-algebras. Math Sem Notes Kobe Univ 10:749–751
Rachůnek J (2001) A duality between algebras of basic logic and bounded representable DRl-monoids. Math Bohemica 126:561–569
Rachůnek J, Šalounová D (2006) Truth values on generalizations of some commutative fuzzy structures. Fuzzy Sets Syst 157:3159–3168
Swamy KLN (1965) Dually residuated lattice ordered semigroups. Math Ann 159:105–114
Vychodil V (2006) Truth-depressing hedges and BL-logic. Fuzzy Sets Syst 157:2074–2090
Ward M, Dilworth RP (1939) Residuated lattices. Trans Am Math Soc 45:335–354
Zadeh L (1975) Fuzzy logic and approximate reasoning. Synthese 30:407–428
Author information
Authors and Affiliations
Corresponding author
Additional information
Research is supported by the Research and Development Council of Czech Government via project MSN 6198959214.
Rights and permissions
About this article
Cite this article
Halaš, R., Botur, M. On very true operators on pocrims. Soft Comput 13, 1063–1072 (2009). https://doi.org/10.1007/s00500-008-0379-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-008-0379-8