The Euler characteristic and valuations on MV-algebras. (Q2877071)

The Euler characteristic of a polyhedron is given by the formula \(\chi=V-E+F\) where \(V\) is the number of vertices, \(E\) is the number of edges, and \(F\) is the number of faces of the polyhedron. Euler's polyhedron formula shows that any convex polyhedron has characteristic 2. More generally one can speak of Euler's characteristic of any topological space by taking the alternating sum of the ranks of \(n^{\text{th}}\) singular homology groups, also known as Betti numbers (the characteristic being defined when the Betti numbers are finite and all equal to zero beyond a certain index). Notably, Euler's characteristic is a homotopy invariant.NEWLINENEWLINE Every finitely presented MV-algebra \(A\) can be represented as the algebra of \(\mathbb Z\)-maps over a rational polyhedron \(P_A\). Thanks to this correspondence, finitely presented MV-algebras can be characterised abstractly as the ones that possess a basis. As an extension of the PhD dissertation of the second author, in the article under review the authors show that for every finitely presented MV-algebra \(A\) there is a unique valuation \(E\) that is idempotent (i.e., \(E(a\oplus b)=E(a\vee b)\)) and assigns value 1 to every element of the basis of \(A\). For each \(a\in A\), \(E(a)\) coincides with the Euler characteristic of the complement in \(P_A\) of the zero-set of \(a\).