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Transactions of the American Mathematical Society

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Isomorphism universal varieties of Heyting algebras

Authors: M. E. Adams, V. Koubek and J. Sichler
Journal: Trans. Amer. Math. Soc. 319 (1990), 309-328
MSC: Primary 06D20; Secondary 03G25, 08A35, 18B15
MathSciNet review: 955486
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Abstract: A variety $ \mathbf{V}$ is group universal if every group $ G$ is isomorphic to the automorphism group $ {\operatorname{Aut}}(A)$ of an algebra $ A \in \mathbf{V}$; if, in addition, all finite groups are thus representable by finite algebras from $ \mathbf{V}$, the variety $ \mathbf{V}$ is said to be finitely group universal. We show that finitely group universal varieties of Heyting algebras are exactly the varieties which are not generated by chains, and that a chain-generated variety $ \mathbf{V}$ is group universal just when it contains a four-element chain. Furthermore, we show that a variety $ \mathbf{V}$ of Heyting algebras is group universal whenever the cyclic group of order three occurs as the automorphism group of some $ A \in \mathbf{V}$. The results are sharp in the sense that, for every group universal variety and for every group $ G$, there is a proper class of pairwise nonisomorphic Heyting algebras $ A \in \mathbf{V}$ for which $ {\operatorname{Aut}}(A) \cong G$.

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