Bases for the positive cone of a partially ordered module

Abstract:

$(R,{R^ + })$ is a partially ordered ring and $(M,{M^ + })$ is a strict $(R,{R^ + })$-module. So M is a left R-module and $({R^ + }\backslash \{ 0\} )({M^ + }\backslash \{ 0\} ) \subseteq {M^ + }\backslash \{ 0\}$. Let $\leqslant ’$ be the order induced on M by ${M^ + }.B \subseteq {M^ + }$ is an ${R^ + }$-basis for ${M^ + }$ means ${R^ + }B = {M^ + }$ (spanning) and if r is in R, b in B with $0 < ’rb \leqslant ’b$ then $rb \notin {R^ + }(B\backslash \{ b\} )$ (independence). Result: If B and D are ${R^ + }$-bases for ${M^ + }$ then card $B =$ card D and to within a permutation ${b_i} = {u_i}{d_i}$, for units ${u_i}$ of ${R^ + }$.