  We study the meet irreducible ideals in certain direct limit algebras, namely
the strongly maximal triangular subalgebras of AF C*-algebras. These ideals
have a description in terms of the coordinates, or spectrum, that is a natural
extension of one description of meet irreducible ideals in the upper triangular
matrices. Additional information is available if the limit algebra is an
analytic subalgebra of its C*-envelope or if the analytic algebra is trivially
analytic with an injective 0-cocycle. Completely meet irreducible ideals are
considered, and a distance formula is presented.
