  A split extension of monoids with kernel k: N -> G, cokernel e: G -> H and
splitting s: H -> G is weakly Schreier if each element g in G can be written g
= k(n)se(g) for some n in N. The characterization of weakly Schreier extensions
allows them to be viewed as something akin to a weak semidirect product. The
motivating examples of such extensions are the Artin glueings of topological
spaces and, of course, the Schreier extensions of monoids which they
generalise. In this paper we show that the lambda-semidirect products of
inverse monoids are also examples of weakly Schreier extensions. The
characterization of weakly Schreier extensions sheds some light on the
structure of lambda-semidirect products. The set of weakly Schreier extensions
between two monoids comes equipped with a natural poset structure, which
induces an order on the set of lambda-semidirect products between two inverse
monoids. We show that Artin glueings are in fact lambda-semidirect products and
inspired by this identify a class of Artin-like lambda-semidirect products. We
show that joins exist for this special class of lambda-semidirect product in
the aforementioned order.
