Euler Characteristic in Gödel and Nilpotent Minimum Logics

Some decades ago, V. Klee and G.-C.Rota introduced a lattice-theoretic analogue of the Euler characteristic, the celebrated topological invariant of polyhedra. In [1], using the Klee-Rota definition, we introduce the Euler characteristic of a formula in Gödel logic, the extension of intuitionistic logic via the prelinearity axiom. We then prove that the Euler characteristic of a formula over n propositional variables coincides with the number of Boolean assignments to these n variables that satisfy the formula. Building on this, we generalise this notion to other invariants that provide additional information about the satisfiability of a formula in Gödel logic. Specifically, the Euler characteristic does not determine non-classical tautologies: the maximum value of the characteristic of a formula over n variables is 2^n, and this can be attained even when the formula is not a tautology in Gödel logic. By contrast, we prove that these new invariants do.