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We develop dimension theory for a large class of structures called espaliers,consisting of a set $L$ equipped with a partial order $\leq$, an orthogonalityrelation $\perp$, and an equivalence relation $\sim$, subject to certainaxioms. The dimension range of $L$ is the universal $\sim$-invarianthomomorphism from $(L,\oplus,0)$ to a partial commutative monoid $S$, where$\oplus$ denotes orthogonal sum in $L$. Particular examples of espaliersinclude (i) complete Boolean algebras, (ii) direct summand lattices ofnonsingular injective modules, (iii) complete, meet-continuous, complemented,modular lattices, and (iv) projection lattices in AW*-algebras. We prove thatthe dimension range of any espalier is a lower interval of a commutative monoidof continuous functions of the form $C(\Omega_{I},Z_\gamma) \timesC(\Omega_{II},R_\gamma) \times C(\Omega_{III},2_\gamma)$, where $\gamma$ is anordinal 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 everylower interval of a monoid of the above form can be represented as thedimension 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 ofthe isomorphism classes of finitely generated projective modules over a ring$R$.

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