  Larsen has recently extended Exel's construction of crossed products from
single endomorphisms to abelian semigroups of endomorphisms, and here we study
two families of her crossed products. First, we look at the natural action of
the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group
$\Gamma$, and the induced action on $C(\Gamma)$. We prove a uniqueness theorem
for the crossed product, and we find a class of connected compact abelian
groups $\Gamma$ for which the crossed product is purely infinite simple.
Second, we consider some natural actions of the additive semigroup
$\mathbb{N}^2$ on the UHF cores in 2-graph algebras, as introduced by Yang, and
confirm that these actions have properties similar to those of single
endomorphisms of the core in Cuntz algebras.
