The Euler characteristic of a formula in G\uf6del logic

Using the lattice-theoretic version of the Euler characteristic introduced by V. Klee and G.-C. Rota, we define the Euler characteristic of a formula in G\uf6del logic (over finitely or infinitely many truth-values). We then prove that the information encoded by the Euler characteristic is classical, i.e., coincides with the analogous notion defined over Boolean logic. Building on this, we define k-valued versions of the Euler characteristic of a formula \u3c6, for each integer k>=2, and prove that they indeed provide information about the logical status of \u3c6 in G\uf6del k-valued logic. Specifically, our main result shows that the k-valued Euler characteristic is an invariant that separates k-valued tautologies from non-tautologies