Ideals in triangular AF algebras

We investigate some structural aspects of the ideal lattice of those triangular AF algebras which arise as inductive limits of triangular matrix algebras. The main class we consider are the join-irreducible ideals, i.e., those ideals {dollar}{lcub}cal I{rcub}{dollar} such that if {dollar}{lcub}cal I{rcub}{dollar} = {dollar}{lcub}cal K{rcub} vee {lcub}cal J{rcub}{dollar} for ideals {dollar}{lcub}cal K{rcub}{dollar} and {dollar}{lcub}cal J{rcub}{dollar}, then either {dollar}{lcub}cal K{rcub}{dollar} = {dollar}{lcub}cal I{rcub}{dollar} or {dollar}{lcub}cal J{rcub}{dollar} = {dollar}{lcub}cal I{rcub}{dollar}. It is shown that semisimple triangular UHF algebras have no join-irreducible ideals and triangular UHF nest algebras admit a large, tractable class of them. The general case is more varied: in some such algebras there are no join-irreducible ideals but in others there are many, depending on the precise manner in which the inductive limit is formed. Furthermore, we show that in many algebras, including the refinement algebra, no ideal of the form P {dollar}{lcub}cal A{rcub}{dollar} {dollar}Qspperp{dollar} for P and Q in Lat {dollar}{lcub}cal A{rcub}{dollar} is join-irreducible. This is in contrast to the case of {dollar}w*{dollar}-closed ideals in nest algebras, where such ideals are the only ones which are join-irreducible.