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monounary_algebras [2012/07/09 13:12]
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monounary_algebras [2012/07/09 13:12] (current)
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Let $j>k\ge 0$ and $m>n\ge 0$. Then Let $j>k\ge 0$ and $m>n\ge 0$. Then
-$\text{Mod}(f^j(x)=f^k(x)\subseteq\text{Mod}(f^m(x)=f^n(x)$ if and only if $k\le n$ and $(j-k)|m-n$.+$\text{Mod}(f^j(x)=f^k(x)\subseteq\text{Mod}(f^m(x)=f^n(x)$ if and only if $k\le n$ and $(j-k)|(m-n)$.
Hence the lattice of nontrivial subvarieties of monounary algebras is isomorphic to $(\mathbb N,\le)\times (\mathbb N,|)$, which is itself isomorphic to the lattice of divisibility of the natural numbers. The variety $\text{Mod}(x=y)$ of trivial subvarieties is the unique element below the variety $\text{Mod}(f(x)=x)$ (which is term-equivalent to the variety of sets). Hence the lattice of nontrivial subvarieties of monounary algebras is isomorphic to $(\mathbb N,\le)\times (\mathbb N,|)$, which is itself isomorphic to the lattice of divisibility of the natural numbers. The variety $\text{Mod}(x=y)$ of trivial subvarieties is the unique element below the variety $\text{Mod}(f(x)=x)$ (which is term-equivalent to the variety of sets).