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| Title of object: |
join |
| Canonical Name: |
Join |
| Type: |
Definition |
| Created on: |
2002-02-24 17:12:12 |
| Modified on: |
2004-02-16 04:48:47 |
| Classification: |
msc:06-XX |
| Defines: |
join-semilattice, join semilattice, upper semilattice |
| Synonyms: |
join=or operator |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts} |
Content:
Certain posets $X$ have a binary operation \emph{join}, denoted $\lor$ such that $x \lor y$ is the least upper bound of $x$ and $y$. Such posets are called \emph{join-semilattices}, or \emph{$\lor$-semilattices}, or \emph{upper semilattices}.
If $j$ and $j'$ are both joins of $x$ and $y$, then $j \leq j'$ and $j' \leq j$, and so $j = j'$; thus a join, if it exists, is unique. The join is also known as the \emph{or operator}. |
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