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

We develop dimension theory for a large class of structures called espaliers, consisting of a set $L$ equipped with a partial order $\le$, 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 ${\mathrm{\Omega }}_{\ast }$ are complete Boolean spaces, and where $Z_{\gamma }$, $R_{\gamma }$, $2_{\gamma }$, respectively, denote the unions of the interval $\{{\mathrm{\aleph }}_{\xi }\mid 0\le \xi \le \gamma \}$ 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 $R$.