join Join 2002-02-24 17:12:12 2005-02-26 05:18:57 Definition join-semilattice join semilattice upper semilattice \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} Certain posets $X$ have a binary operation \emph{join} denoted by $\lor$, 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}.