The complete dimension theory of partially ordered systems with equivalence and orthogonality
| dc.creator | Goodearl, K. R. | |
| dc.creator | Wehrung, F. | |
| dc.date | 2004-03-02 | |
| dc.date.accessioned | 2026-07-07T05:05:53Z | |
| dc.date.available | 2026-07-07T05:05:53Z | |
| dc.description | 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 $C(Ω_{I},Z_γ) \times C(Ω_{II},R_γ) \times C(Ω_{III},2_γ)$, where $γ$ is an ordinal and the $Ω_{*}$ are complete Boolean spaces, and where $Z_γ$, $R_γ$, $2_γ$, respectively, denote the unions of the interval $\{\aleph_ξ\mid 0\le ξ\le γ\}$ 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 $R$. | |
| dc.description | latex file with index file | |
| dc.identifier | https://arxiv.org/abs/math/0403057 | |
| dc.identifier | http://arxiv.org/abs/math/0403057 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70338 | |
| dc.subject | General Mathematics | |
| dc.subject | Operator Algebras | |
| dc.subject | Rings and Algebras | |
| dc.subject | 06B15, 06C15, 16D50, 16D70, 46L10 | |
| dc.title | The complete dimension theory of partially ordered systems with equivalence and orthogonality | |
| dc.type | text |