Valuations in Nilpotent Minimum Logic

The Euler characteristic can be defined as a special kind of valuation on finite distributive lattices. This work begins with some brief consideration on the role of the Euler characteristic on NM algebras, the algebraic counterpart of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified version of the Euler characteristic we call idempotent Euler characteristic. We show that the new valuation encodes information about the formul{\ae} in NM propositional logic

On the independent subsets of powers of paths and cycles

In the first part of this work we provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself. In the second part we consider the case of cycles. We evaluate the number of edges of the Hasse diagram of the independent subsets of the h-th power of a cycle ordered by inclusion. For h=1, and n>1, such a value is the number of edges of a Lucas cube.Comment: 9 pages, 4 figure

Eulerian digraphs and Dyck words, a bijection

The main goal of this work is to establish a bijection between Dyck words and a family of Eulerian digraphs. We do so by providing two algorithms implementing such bijection in both directions. The connection between Dyck words and Eulerian digraphs exploits a novel combinatorial structure: a binary matrix, we call Dyck matrix, representing the cycles of an Eulerian digraph

Making simple proofs simpler

An open partition \pi{} [Cod09a, Cod09b] of a tree T is a partition of the vertices of T with the property that, for each block B of \pi, the upset of B is a union of blocks of \pi. This paper deals with the number, NP(n), of open partitions of the tree, V_n, made of two chains with n points each, that share the root