  In Archimedean vector lattices bands can be introduced via three different
coinciding notions. First, they are order closed ideals. Second, they are
precisely those ideals which equal their double disjoint complements. The third
concept is that of an ideal which contains the supremum of any of its bounded
subsets, provided the supremum exists in the vector lattice. We investigate
these three notions and their relationships in the more general setting of
Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed
ideal, which is related to the third aforementioned notion in vector lattices.
We show that for a directed ideal $I$ in a pervasive pre-Riesz space with the
Riesz decomposition property these three concepts coincide, provided the double
disjoint complement of $I$ is directed. In pervasive pre-Riesz spaces every
directed band is supremum closed and every supremum closed directed ideal $I$
equals its double disjoint complement, provided the double disjoint complement
of $I$ is directed. In general, in Archimedean pre-Riesz spaces the three
notions differ. For this we provide appropriate counterexamples.
