We introduce the concept of very true operator on a commutative basic algebra in a way analogous to that for fuzzy logics. We are motivated by the fact that commutative basic algebras form an algebraic axiomatization of certain non-associative fuzzy logics. We prove that every such operator is fully determined by a certain relatively complete sublattice provided its idempotency is assumed.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1220,
author = {Ivan Chajda and Filip \v Svr\v cek},
title = {Sublattices corresponding to very true operators in commutative basic algebras},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {34},
year = {2014},
pages = {183-189},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1220}
}
Ivan Chajda; Filip Švrček. Sublattices corresponding to very true operators in commutative basic algebras. Discussiones Mathematicae - General Algebra and Applications, Tome 34 (2014) pp. 183-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1220/
[000] [1] M. Botur and F. Švrček, Very true on CBA fuzzy logic, Math. Slovaca 60 (2010) 435-446. doi: 10.2478/s12175-010-0023-9. | Zbl 1240.06043
[001] [2] M. Botur, I. Chajda and R. Halaš, Are basic algebras residuated lattices?, Soft Comp. 14 (2010) 251-255. doi: 10.1007/s00500-009-0399-z. | Zbl 1188.03048
[002] [3] I. Chajda, R. Halaš and J. Kühr, Distributive lattices with sectionally antitone involutions, Acta Sci. Math. (Szeged) 71 (2005) 19-33. | Zbl 1099.06006
[003] [4] I. Chajda, R. Halaš and J. Kühr, Semilattice Structures (Heldermann Verlag (Lemgo, Germany), 2007).
[004] [5] P. Hájek, On very true, Fuzzy Sets and Systems 124 (2001) 329-333. | Zbl 0997.03028
[005] [6] L.A. Zadeh, A fuzzy-set-theoretical interpretation of linguistic hedges, J. Cybern. 2 (1972) 4-34.