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

Abstract

We develop dimension theory for a large class of structures called espaliers, consisting of a set $L$ equipped with a partial order $\leq$, an orthogonality relation $\perp$, and an equivalence relation $\sim$, subject to certain axioms. The dimension range of $L$ is the universal $\sim$-invariant homomorphism from $(L,\oplus,0)$ to a partial commutative monoid $S$, where $\oplus$ denotes orthogonal sum in $L$. Particular examples of espaliers include (i) complete Boolean algebras, (ii) direct summand lattices of nonsingular injective modules, (iii) complete, meet-continuous, complemented, modular lattices, and (iv) projection lattices in AW*-algebras. We prove that the dimension range of any espalier is a lower interval of a commutative monoid of continuous functions of the form $C(\Omega_{I},Z_\gamma) \times C(\Omega_{II},R_\gamma) \times C(\Omega_{III},2_\gamma)$, where $\gamma$ is an ordinal and the $\Omega_{*}$ are complete Boolean spaces, and where $Z_\gamma$, $R_\gamma$, $2_\gamma$, respectively, denote the unions of the interval $\{\aleph_\xi \mid 0\le \xi\le \gamma\}$ with the sets of nonnegative integers, nonnegative real numbers, and 0, respectively. Conversely, we prove that every lower interval of a monoid of the above form can be represented as the dimension range of an espalier arising from each of the contexts (i)--(iv) above. As corollaries in cases (ii) and (iv), we obtain complete descriptions (both function-theoretic and axiomatic) of the monoids $V(R)$, consisting of the isomorphism classes of finitely generated projective modules over a ring $R$.