  R.M. Green described structural properties that ``doubly infinite'' colored
posets should possess so that they can be used to construct representations of
most affine Kac--Moody algebras. These representations are analogs of the
minuscule representations of the semisimple Lie algebras, and his posets
(``full heaps'') are analogs of the finite minuscule posets. Here only simply
laced Kac--Moody algebras are considered. Working with their derived
subalgebras, we provide a converse to Green's theorem. Smaller collections of
colored structural properties are also shown to be necessary and sufficient for
such poset-built representations to be produced for smaller subalgebras,
especially the ``Borel derived'' subalgebra. These developments lead to the
formulation of unified definitions of finite and infinite colored minuscule and
$d$-complete posets. This paper launches a program that seeks to extend the
notion of ``minuscule representation'' to Kac--Moody algebras, and to classify
such representations.
