  We investigate which weighted convolution algebras $\ell^1_\omega(S)$, where
$S$ is a semilattice, are AMNM in the sense of Johnson (JLMS, 1986). We give an
explicit example where this is not the case. We show that the unweighted
examples are all AMNM, as are all $\ell^1_\omega(S)$ where $S$ has either
finite width or finite height. Some of these finite-width examples are
isomorphic to function algebras studied by Feinstein (IJMMS, 1999).
  We also investigate when $(\ell^1_\omega(S),{\bf M}_2)$ is an AMNM pair in
the sense of Johnson (JLMS, 1988), where ${\bf M}_2$ denotes the algebra of
2-by-2 complex matrices. In particular, we obtain the following two contrasting
results: (i) for many non-trivial weights on the totally ordered semilattice
${\bf N}_{\min}$, the pair $(\ell^1_\omega({\bf N}_{\min}),{\bf M}_2)$ is not
AMNM; (ii) for any semilattice $S$, the pair $(\ell^1(S),{\bf M}_2)$ is AMNM.
The latter result requires a detailed analysis of approximately commuting,
approximately idempotent $2\times 2$ matrices.
