Abstract
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.
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This research was supported by a grant of National Natural Science Foundation of China (11571281).
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Wang, W., Xin, X.L. & Wang, J.T. EQ-algebras with internal states. Soft Comput 22, 2825–2841 (2018). https://doi.org/10.1007/s00500-017-2754-9
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DOI: https://doi.org/10.1007/s00500-017-2754-9