# Differences

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varieties [2010/07/29 17:08]
jipsen created
varieties [2010/08/20 10:08] (current)
jipsen
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Varieties are also called \emph{equational classes}. Varieties are also called \emph{equational classes}.
-By a fundamental result of [(Garrett Birkhoff, <i>On the structure of abstract algebras</i>, +By a fundamental result of Birkhoff[(Garrett Birkhoff, \emph{On the structure of abstract algebras},
Proceedings of the Cambridge Philosophical Society, 31:433--454, 1935)] a class $\mathcal{K}$ of algebras is a Proceedings of the Cambridge Philosophical Society, 31:433--454, 1935)] a class $\mathcal{K}$ of algebras is a
variety iff it is closed under the operators $H$, $S$, $P$ (i.e., $H\mathcal{K}\subseteq\mathcal{K}$, $S\mathcal{K}\subseteq\mathcal{K}$, and $P\mathcal{K}\subseteq\mathcal{K}$), where variety iff it is closed under the operators $H$, $S$, $P$ (i.e., $H\mathcal{K}\subseteq\mathcal{K}$, $S\mathcal{K}\subseteq\mathcal{K}$, and $P\mathcal{K}\subseteq\mathcal{K}$), where
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$P\mathcal{K}=\{$direct products of members of $\mathcal{K}\}$. $P\mathcal{K}=\{$direct products of members of $\mathcal{K}\}$.
-See [http://www.thoralf.uwaterloo.ca/htdocs/ualg.html Stanley N. Burris and H.P. Sankappanavar,  A Course in Universal Algebra] for+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
more details. more details.
-Show all pages on [http://math.chapman.edu/cgi-bin/structures.pl?dosearch=1&search=value&gt;variety varieties]+Show all pages on [[http://math.chapman.edu/~jipsen/structures/doku.php/?do=search&id=variety&amp;fulltext=Search| varieties]]
-A picture of some [http://www.chapman.edu/~jipsen/PCP/theoriesPO1.html theories ordered by interpretability]+A picture of some [[http://www.chapman.edu/~jipsen/PCP/theoriesPO1.html| theories ordered by interpretability]]
=== Some varieties and quasivarieties listed by signature and (first) subclass relation === === Some varieties and quasivarieties listed by signature and (first) subclass relation ===