# Differences

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

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>variety varieties] | + | Show all pages on [[http://math.chapman.edu/~jipsen/structures/doku.php/?do=search&id=variety&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 === |

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