Soft Computing

, Volume 22, Issue 9, pp 2825–2841 | Cite as

EQ-algebras with internal states

  • Wei Wang
  • Xiao Long Xin
  • Jun Tao Wang


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.


SEQ-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.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China

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