  We study the interplay between three weak topologies on a topological
semilattice $X$: the weak$^\circ$ topology $\mathcal W^\circ_X$ (generated by
the base consiting of open subsemilattices of $X$), the weak$^\bullet$ topology
$\mathcal W^\bullet_X$ (generated by the subbase consisting of complements to
closed subsemilattices), and the $\mathbb I$-weak topology $\mathcal W_X$
(which is the weakest topology in which all continuous homomorphisms $h:X\to
[0,1]$ remain continuous). Also we study the interplay between the weak
topologies $\mathcal W^\bullet_X$, $\mathcal W^\circ_X$, $\mathcal W_X$ of a
topological semilattice $X$ and the Scott and Lawson topologies $\mathcal S_X$
and $\mathcal L_X$, which are determined by the order structure of the
semilattice.
  We prove that the weak$^\bullet$ topology $\mathcal W^\bullet$ on a Hausdorff
semitopological semilattice $X$ is compact if and only if $X$ is chain-compact
in the sense that each closed chain in $X$ is compact. This result implies that
the Lawson topology $\mathcal L_X$ on a semilattice $X$ is compact if and only
if $X$ is a continuous semilattice if and only if $X$ complete in the sense
that each non-empty chain $C$ in $X$ has $\inf(C)$ and $\sup(C)$ in $X$.
  For a chain-compact Hausdorff topological semilattice $X$ with topology
$\mathcal T_X$ we prove the inclusions $\mathcal W_X\subset\mathcal
L_X\subset\mathcal W^\bullet_X\subset\mathcal T_X$. For a compact topological
semilattice $X$ we prove that $\mathcal T_X=\mathcal W^\bullet_X$ if and only
if $\mathcal T_X=\mathcal L_X$ if and only if $\mathcal T_X=\mathcal L_X$.
