I don´t understand the definition of discrete invariant and I wonder if someone of you would know it. The notion appears in the following sense:
Given a set $M$ and equivalence relation $\sim$ on $M$, a discrete invariant is a function $f:M/\sim\rightarrow\mathbb{Z}$ that partitions $M/\sim$. What does it mean?
Well, the function $f:M/\equiv \;\rightarrow {\Bbb Z}$ assigns to each equivalence class $\bar a =\{b\in M\mid a\equiv b\}$ an integer $f(\bar a)$.
For instance, take the set of words $M=\{\text{car}, \text{auto},\text{face},\text{book},\text{app}\}$. If the equivalence relation is ''same number of letters as'', then $$M/\equiv = \{\{\text{car},\text{app}\},\{\text{auto},\text{face},\text{book}\}\}$$ and the invariant $f$ could assign the ''length information'' to each class: $f(\{\text{car},\text{app}\}) = 3$ and $f(\{\text{auto},\text{face},\text{book}\})=4$.