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2006•Bulletin of the Australian Mathematical SocietyOpen access

Cyclic Tarski algebras

Marta A. Zander

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Abstract

The variety of cyclic Boolean algebras is a particular subvariety of the variety of tense algebras. The objective of this paper is to study the variety  of {→,g, h}-subreducts of cyclic Boolean algebras, which we call cyclic Tarski algebras. We prove that  is generated by its finite members and we characterise the locally finite subvarieties of . We prove that there are no splitting varieties in the lattice Λ() of subvarieties of . Finally, we prove that the subquasivarieties and the subvarieties of a locally finite subvariety of  coincide.

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What this paper is about

The variety of cyclic Boolean algebras is a particular subvariety of the variety of tense algebras. The objective of this paper is to study the variety  of {→,g, h}-subreducts of cyclic Boolean algebras, which we call cyclic Tarski algebras. We prove that  is generated by its finite members and we characterise the locally finite subvarieties of . We prove that there are no splitting varieties in the lattice Λ() of subvarieties of . Finally, we prove that the subquasivarieties and the subvarieties of a locally finite subvariety of  coincide.

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Available abstract

The variety of cyclic Boolean algebras is a particular subvariety of the variety of tense algebras. The objective of this paper is to study the variety  of {→,g, h}-subreducts of cyclic Boolean algebras, which we call cyclic Tarski algebras. We prove that  is generated by its finite members and we characterise the locally finite subvarieties of . We prove that there are no splitting varieties in the lattice Λ() of subvarieties of . Finally, we prove that the subquasivarieties and the subvarieties of a locally finite subvariety of  coincide.

Key concepts: Subvariety, Mathematics, Variety (cybernetics), Boolean algebras canonically defined, Interior algebra, Pure mathematics, Lattice (music), Stone's representation theorem for Boolean algebras

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