Inverses of disjointness preserving operators in finite dimensional pre-Riesz spaces

Abstract

If ℝⁿ is partially ordered by a generating closed cone K; then (ℝⁿ;K) is a pre-Riesz space. We show for a disjointness preserving bijection T on (ℝⁿ;K) that the inverse of T is also disjointness preserving. We prove that for T there is k ∈ P(b) such that T^k is band preserving, where b is the number of bands in (ℝn;K); and P(b) the set of orders of permutations on b symbols.

Mathematics Subject Classification (2010): 47B60.

Keywords: Band, band preserving operator, disjointness preserving operator, d-isomor-phism, nite dimensional, generating closed cone, pre-Riesz space.