The existence of states on EQ-algebras

Inspired by the open problems “How to define the notions of fantastic filters and states in EQ-algebras” in [LIU, L. Z.—ZHANG, X. Y.: Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Syst. 26 (2014), 2087–2097], we introduce the notions of fantastic filters and investigate the existence of Bosbach states and Riečan states on EQ-algebras by use of fantastic filters. Firstly, we prove that a residuated EQ-algebra has a Bosbach state if and only if it has a fantastic filter. We also establish that a good EQ-algebra has a state-morphism if and only if it has a prime fantastic filter. Furthermore, we introduce the notion of QI-EQ-algebras and obtain the necessary and sufficient condition for a residuated QI-EQ-algebra having Riečan states. Finally, we introduce the notion of semi-divisible EQ-algebras and give an example of a semi-divisible residuated EQ-algebra, which is not a semi-divisible residuated lattice. We also prove that every semi-divisible residuated EQ-algebra admits Riečan states. These works generalize a series of existing results about existence of states in several algebras, such as residuated lattices, NM-algebras, MTL-algebras, BL-algebras and so on.

Folding Theory for Fantastic Filters in BL-Algebras

In this paper, the author examines the notion of n-fold fantastic and fuzzy n-fold fantastic filters in BL-algebras. Several characterizations of fuzzy n-fold fantastic filters are given. The author shows that every n-fold (fuzzy n-fold) fantastic filter is a filter (fuzzy filter), but the converse is not true. Using a level set of a fuzzy set in a BL-algebra, the author gives a characterization of fuzzy n-fold fantastic filters. Finally, the author establishes the extension property for n-fold and fuzzy n-fold fantastic filters in BL-algebras. The author also constructs some algorithms for folding theory applied to fantastic filters in BL-algebras.

Folding Theory for Fantastic Filters in BL-Algebras

In this paper, the author examines the notion of n-fold fantastic and fuzzy n-fold fantastic filters in BL-algebras. Several characterizations of fuzzy n-fold fantastic filters are given. The author shows that every n-fold (fuzzy n-fold) fantastic filter is a filter (fuzzy filter), but the converse is not true. Using a level set of a fuzzy set in a BL-algebra, the author gives a characterization of fuzzy n-fold fantastic filters. Finally, the author establishes the extension property for n-fold and fuzzy n-fold fantastic filters in BL-algebras. The author also constructs some algorithms for folding theory applied to fantastic filters in BL-algebras.

Fantastic filters of lattice implication algebras

The notion of a fantastic filter in a lattice implication algebra is introduced, and the relations among filter, positive implicative filter, and fantastic filter are given. We investigate an equivalent condition for a filter to be fantastic, and state an extension property for fantastic filter.