States on Orthomodular Posets of Decompositions

In Harding (1996; see also Harding, 1998, 1999), a method was given to construct an orthomodular poset Fact X from the direct product decompositions of a set. As this forms the basis of our study we briefly review the pertinent facts. For a set X let Eq X denote the set of equivalence relations on X, use ◦ for relational product, 1 for the least equivalence relation on X, and ∇ for the largest. Define Fact X = {(θ, θ ′) | θ, θ ′ ∈ Eq X, θ ∩ θ ′ = 1, θ ◦ θ ′ = ∇}. Let≤ be the relation on Fact X defined by setting (θ, θ ′) ≤ (φ, φ′) if θ ⊆ φ, φ′ ⊆ θ , and all of θ, θ ′, φ, φ′ permute under relational product. Define a unary operation ⊥ on Fact X by setting (θ, θ ′)⊥ = (θ ′, θ ). Then as shown in Harding (1996), (Fact X,≤,⊥) is an orthomodular poset with (θ, θ ′) ∨ (φ, φ′) = (θ ◦ φ, θ ′ ∩ φ′) for (θ, θ ′) orthogonal to (φ, φ′). The notation Fact X is used because such pairs (θ, θ ′) are commonly called factor pairs, and Fact X is called the orthomodular poset of decompositions of X because the factor pairs are exactly those pairs of equivalence relations that occur as the kernels of the projection operators associated with a binary direct product decomposition X ∼= Y × Z . If X is the underlying set of some algebra A, then the factor pairs (θ, θ ′) which are compatible with this additional structure (i.e., which are congruences) form a suborthomodular poset Fact A of Fact X. For a vector space V the correspondence between subspaces of V and congruences of V, as well as the fact that all