Connections between GE algebras and pre-Hilbert algebras

Document Type : Research Paper

Author

Institute of Mathematics, Faculty of Exact and Natural Science, Siedlce University, PL-08110 Siedlce, Poland

Abstract

GE algebras (generalized exchange algebras) and pre-Hilbert algebras are a generalization of well-known Hilbert algebras. In the paper, connections between these algebras are studied. In particular, it is proven that pi-BE algebras are equivalent with GE algebras satisfying the exchange property. Some characterizations of transitive GE algebras and exchange pre-Hilbert algebras are given. It is shown that the intersection of classes of GE algebras and pre-Hilbert algebras is the class of transitive GE algebras. Moreover, GE, BE and pre-Hilbert algebras with the antisymmetry property are investigated. It is proven that transitive GE algebras satisfying the property of antisymmetry coincide with Hilbert algebras. Finally, positive implicative GE and pre-Hilbert algebras are considered, their connections with some algebras of logic are presented. In addition, the hiearchies existing between the classes of algebras studied here are shown.

Keywords

References

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Algebraic Structures and Their Applications

Articles in Press, Accepted Manuscript
Available Online from 27 November 2023

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