The complete dimension theory of partially ordered systems with equivalence and orthogonality

We develop dimension theory for a large class of structures of the form (L,≤,⊥,∼), where (L ≤) is a partially ordered set, ⊥ is a binary relation on L, and ∼ is an equivalence relation on L, subject to certain axioms. We call these structures espaliers. For x, y, z ∈ L, we say that z = x ⊕ y holds, if x ⊥ y and z is the supremum of {x, y}. The dimension theory of L is the universal ∼-invariant homomorphism from (L,⊕, 0) to a partial commutative monoid S. We say that S is the dimension range of L. Particular examples of espaliers are the following: (i) Let B be a complete Boolean algebra. For x, y ∈ B, we say that x ⊥ y if x ∧ y = 0, and we take ∼ to be any zero-separating, unrestrictedly additive and refining equivalence relation on B (for instance, equality). (ii) Let R be a right self-injective von Neumann regular ring. We denote by L the lattice of all direct summands of a given nonsingular injective right R-module, for instance, the lattice of finitely generated right ideals of R. For A, B ∈ L, we say that A ⊥ B if A ∩ B = {0}, and A ∼ B if A ∼= B. (iii) More generally, let L be a complete, meet-continuous, complemented, modular lattice. For x, y ∈ L, we say that x ⊥ y if x ∧ y = 0, and x ∼ y if x and y are projective by (finite) decomposition. (iv) Let A be an AW*-algebra. We denote by L the lattice of projections of A, and take the standard orthogonality and equivalence relations on L. For p, q ∈ L, then, p ⊥ q if pq = 0, and p ∼ q if p and q are Murrayvon Neumann equivalent, that is, there exists x ∈ A such that p = xx and q = xx. We prove that the dimension range of any espalier (L,≤,⊥,∼) is a lower interval of a commutative monoid of the form C(ΩI,Zγ)× C(ΩII,Rγ)× C(ΩIII, 2γ), (*) where ΩI, ΩII, and ΩIII are complete Boolean spaces, and where we put, for every ordinal γ, Zγ = Z ∪ {אξ | 0 ≤ ξ ≤ γ}, Rγ = R ∪ {אξ | 0 ≤ ξ ≤ γ}, 2γ = {0} ∪ {אξ | 0 ≤ ξ ≤ γ}, endowed with their interval topology and natural addition operations. Conversely, we prove that every lower interval of a monoid of the form (*) can be represented as the dimension range of an espalier arising from each of the contexts (i)–(iv) above. The context of W*-algebras requires the spaces ΩI, ΩII, and ΩIII to be hyperstonian, and no further restriction is needed. This subsumes many earlier dimension-theoretic results, and, in applications, completes theories developed for examples such as (i)–(iv) above.