Krull Dimension and Gabriel Dimension of an Ordered Set

Abstract

For a partially ordered set (L,≤) we let Г(L) be the set {(a,b), a ≤ b, a,b ∈ L}. By transfinite recursion we may define on Г(L) a filtration in the following way:

• Г_-1(L) = {(a,b) ∈ Г(L),a = b}

• Г₀(L) = {(a,b) ∈ Г(L), [a,b] is Artinian}

Г_α(L) = {(a,b) ∈ Г(L), for all b ≥ b₁ ≥… ≥ b _n ≥… ≥ a, there is an n ∈ ℕ such that [b_i+1, b _i ) ∈ U_β<α Г _β (L), for each i ≥ n}, where we assume that for each ordinal β, β < α, Г _β (L) is already defined. In this way we obtain an ascending chain Г_-1(L) ⊂ Г₀(L) ⊂… ⊂ Г _α (L) ⊂.... Since L is a set it follows that Г_ς(L) = Г_ς+1(L) =…. If Г(L) = Г _α (L) for some ordinal α then we say that the Krull dimension of L is defined.