The uniform dimension of a monoid with applications to graph algebras

We adapt Goldie's concept of uniform dimensions from module theory over rings to Γ-monoids. A Γ-monoid M is said to have uniform dimension n if n is the largest number of pairwise incomparable nonzero Γ-order ideals contained in M. Specializing to the talented monoid of a graph, we show that the uniform dimension provides a rough measure of how the graph branches out. Since for any order ideal I, its orthogonal ideal I^⊥ is the largest ideal incomparable to I, we study the notions of orthogonality and regularity, particularly when I^⊥⊥=I. We show that the freeness of the action of Z on the talented monoid of a graph is preserved under quotienting by a regular ideal. Furthermore, we determine the underlying hereditary and saturated sets that generate these ideals. These results unify recent studies on regular ideals of the corresponding Leavitt path algebras and graph C^⁎-algebras. We conclude that for row-finite graphs E and F, if there is a Z-monoid isomorphism T_E≅T_F, then there is a one-to-one correspondence between the regular ideals of the associated Leavitt path algebras L_k(E) and L_k(F) (and similarly, C^⁎(E) and C^⁎(F)). Since the talented monoid T_E is the positive cone of the graded Grothendieck group K₀^gr(L_k(E)), this provides further evidence supporting the Graded Classification Conjecture.