Classification of compatible module orderings (Q1805913)

Let \(R\) be the polynomial ring over the field \(k\) in the indeterminates \(x_1,\dots,x_n\) and let \(M\) be any finitely generated free module over \(R\). Put \(T^n=\{x_1^{a_1} \dots x_n^{a_n} \mid a_i\in \mathbb{N}\}\) and \(MT(M)= \{te_i\mid t\in T^n\}\), where \(\{e_i\}_{i\in S}\) is the set of unitary vectors of \(M\). If \(\tau\) denotes any total ordering on \(T^n\), then a total order relation \(\mu\) on \(MT(M)\) is called a Riquier ordering if it satisfies certain compatibility conditions with respect to \(\tau\). The ain result of the paper gives a classification of all total order relations on \(M\) by means of the Riquier orderings on \(MT(M)\) and an isotone embedding of \((M,\mu)\) into an ordered polynomial ring over \(k\) in \(x_1,\dots,x_n\) and some further indeterminates.