EQ-algebras with internal states
- 181 Downloads
The main goal of this paper is to investigate EQ-algebras with internal states and state morphism good EQ-algebras. To begin with, we introduce the notion of EQ-algebras with internal states (simplify, SEQ-algebras) and discuss the relation between SEQ-algebras and state EQ-algebras. In the following, we study state filters (simplify, S-filters) and state prefilters (simplify, S-prefilters) of SEQ-algebras and discuss subdirectly irreducible SEQ-algebras. We focus on algebraic structures of the set SPF\((E,\sigma )\) of all S-prefilters on a SEQ-algebra and obtain that SPF\((E,\sigma )\) forms a complete Brouwerian lattice, when E is an \(\ell \)EQ-algebra or good. Moreover, for \(\ell \)EQ-algebras, SPF\((E,\sigma )\) forms a Heyting algebra if \(\sigma \) is faithful and preserves \(\rightarrow \). Then, we introduce the \(\sigma \)-co-annihilator of a non-empty set A on a SEQ-algebra. As applications, we give a characterization for minimal prime S-prefilters of state morphism good EQ-algebras and characterize the representable state morphism good EQ-algebras by minimal prime S-prefilters.
KeywordsSEQ-algebra S-prefilter S-filter \(\sigma \)-Co-annihilator Representable
This research was supported by a grant of National Natural Science Foundation of China (11571281).
Compliance with ethical standards
Conflict of interest
The authors declare that there is no conflict of interests.
This article does not contain any studies with human participants or animals performed by any of the authors.
Informed consent was obtained from all individual participants included in the study.
- Flaminio T, Montagna F (2007) An algebraic approach to states on MV-algebras. In: Štěpnička M, Novák V, Bodenhofer U (eds) Proceedings of the 5th EUSFLAT conference, Ostrava,Czech Republic, 11–14 Sept, vol 2, pp 201–206Google Scholar
- Novák V (2005b) Fuzzy type theory as higher-order fuzzy logic. In: Proceedings of the 6th international conference on intelligent technologies, Bangkok, ThailandGoogle Scholar
- Novák V (2006) EQ-algebras: primary concepts and properties. In: Proceedings of the Czech–Japan seminar, ninth meeting, Kitakyushu and Nagasaki, 18–22 Aug, Graduate School of Information, Waseda University, pp 219–223Google Scholar
- Riečan B (2000) On the probability on BL-algebras. Acta Math Nitra 4:3–13Google Scholar