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Reports on Mathematical Logic

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No. 32 (1998)

Tomasz KOWALSKI

Varieties of Tense Algebras
A b s t r a c t.
The paper has two parts preceded by
quite comprehensive preliminaries.

In the first part it is shown that a subvariety of the variety ${\cal T}$ of all tense
algebras is discriminator if and only if it is semisimple. The variety ${\cal T}$
turns out to be the join of an increasing chain of varieties ${\cal D}_n$,
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 ${\cal D}_n$ can be further
characterised by certain natural conditions on Kripke frames.

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

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.

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