# Differences

This shows you the differences between two versions of the page.

varieties [2010/07/29 17:23] jipsen |
varieties [2010/08/20 10:08] (current) jipsen |
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$S\mathcal{K}=\{$subalgebras of members of $\mathcal{K}\}$\\ | $S\mathcal{K}=\{$subalgebras of members of $\mathcal{K}\}$\\ | ||

$P\mathcal{K}=\{$direct products of members of $\mathcal{K}\}$. | $P\mathcal{K}=\{$direct products of members of $\mathcal{K}\}$. | ||

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+ | Equivalently, $\mathcal K$ is a variety iff $\mathcal K=HSP\mathcal K$. | ||

+ | |||

+ | In particular, given any class $\mathcal K$ of algebras, $V\mathcal K=HSP\mathcal K$ is the smallest variety that contains $\mathcal K$, and is called the \emph{variety generated by $\mathcal K$}. | ||

See [[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html| Stanley N. Burris and H.P. Sankappanavar, A Course in Universal Algebra]] for | See [[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html| Stanley N. Burris and H.P. Sankappanavar, A Course in Universal Algebra]] for |

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