Generalized deviations of posets (Q1184857)

Let \((T,\leq)\) be a linear order of type \(\tau\). Let \((S,\leq)\) be a partially ordered set and \(a\), \(b\) be elements of \(S\). The author defines \(\text{dev}_ \tau(a,b)\) as follows: (0) \(\text{dev}_ \tau(a,b)=- 1\Leftrightarrow a=b\); (1) \(\text{dev}_ \tau(a,b)=0\Leftrightarrow\) there is no embedding of \(T\) into \([a,b]\); (2) for each ordinal \(\alpha>0\), \(\text{dev}_ \tau(a,b)=\alpha\Leftrightarrow\text{dev}_ \tau(a,b)\not<\alpha\) and for each monotone \(f: T\to[a,b]\) there are \(x<y\) in \(T\) with \(\text{dev}_ \tau(f(x),f(y))<\alpha\). Then the \(\tau\)-deviation of \(S\) is the supremum of \(\{\text{dev}_ \tau(a,b)\); \(a\leq b\) in \(S\}\). (For \(S\) bounded, the notion of \(\tau\)-deviation agrees with the similar notion of \textit{M. Pouzet} and \textit{N. Zaguia} [Discrete Math. 53, 173-192 (1985; Zbl 0579.06002)].) The \(\tau\)- deviation of a poset is a generalized abstraction of the Krull dimension of a module. The paper develops the analysis of the connection between deviation and chain depth and length, and gives a technique for computation.