The existence of states on EQ-algebras

Year, pages: 2020, 527 - 546

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.: \textit{Implicative and positive implicative prefilters of\linebreak EQ-algebras}, J. Intell. Fuzzy Syst. \textbf{26} (2014), 2087–2097], we introduce the notions of fantastic filters and investigate the existence of Bosbach states and Rie\v{c}an 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 Rie\v{c}an 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 Rie\v{c}an 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.

Xin, X., Ma, Y., Fu, Y. 2020. The existence of states on EQ-algebras. In Mathematica Slovaca, vol. 70, no.3, pp. 527-546. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0369

Xin, X., Ma, Y., Fu, Y. (2020). The existence of states on EQ-algebras. Mathematica Slovaca, 70(3), 527-546. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0369

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava