Varieties of tense algebras.

*(English)*Zbl 0941.03066Summary: The paper has two parts preceded by quite comprehensive preliminaries.

In the first part it is shown that a subvariety of the variety \({\mathcal T}\) of all tense algebras is discriminator if and only if it is semisimple. The variety \({\mathcal T}\) turns out to be the join of an increasing chain of varieties \({\mathcal D}_n\), which are discriminator varieties. The argument carries over to all finite type varieties of Boolean algebras with operators satisfying some term conditions. In the case of tense algebras, the varieties \({\mathcal D}_n\) can be further characterised by certain natural conditions on Kripke frames.

In the second part it is shown that the lattice of subvarieties of \({\mathcal D}_0\) has two atoms, the lattice of subvarieties of \({\mathcal D}_1\) has countably many atoms, and for \(n<1\), the lattice of subvarieties of \({\mathcal D}_n\) has continuum atoms. The proof of the second of the above statements involves a rather detailed description of zero-generated simple algebras in \({\mathcal D}_1\).

Almost all the arguments are cast in algebraic form, but both parts begin with an outline describing their contents from the dual point of view of tense logics.

In the first part it is shown that a subvariety of the variety \({\mathcal T}\) of all tense algebras is discriminator if and only if it is semisimple. The variety \({\mathcal T}\) turns out to be the join of an increasing chain of varieties \({\mathcal D}_n\), which are discriminator varieties. The argument carries over to all finite type varieties of Boolean algebras with operators satisfying some term conditions. In the case of tense algebras, the varieties \({\mathcal D}_n\) can be further characterised by certain natural conditions on Kripke frames.

In the second part it is shown that the lattice of subvarieties of \({\mathcal D}_0\) has two atoms, the lattice of subvarieties of \({\mathcal D}_1\) has countably many atoms, and for \(n<1\), the lattice of subvarieties of \({\mathcal D}_n\) has continuum atoms. The proof of the second of the above statements involves a rather detailed description of zero-generated simple algebras in \({\mathcal D}_1\).

Almost all the arguments are cast in algebraic form, but both parts begin with an outline describing their contents from the dual point of view of tense logics.