Isomorphism universal varieties of Heyting algebras

Adams, M. E.; Koubek, V.; Sichler, J.

doi:10.1090/S0002-9947-1990-0955486-9

Isomorphism universal varieties of Heyting algebras
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by M. E. Adams, V. Koubek and J. Sichler PDF

Trans. Amer. Math. Soc. 319 (1990), 309-328 Request permission

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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