We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra I(X, K), where X is a finite poset (star, Y and Rhombuses) and K is a finite field of characteristic different from 2. Using the techniques describing here we show an algorithm to calculate the number of involutions on any finite poset.
The main objective of this paper is to prove Monsky's Theorem, that provides a beautiful application of the 2-adic valuation in order to solve a plane geometry problem. This theorem states that given any dissection of a square into finitely many nonoverlapping triangles of equal area the number of triangles must be even. In order to prove this statement, we will need some previous results from Combinatorial Topology and Algebra.Keywords: Dissection of a square into triangles of equal area, 2-adic valuation.
RESUMOO objetivo principal desse trabalho é a demonstração do Teorema de Monsky, o qual fornece uma bela aplicação da valoração 2-ádica na resolução de um problema de Geometria Plana. Esse teorema afirma que dada qualquer dissecção de um quadrado em triângulos não sobrepostos e de mesma área, o número de triângulos deve ser par. Com o objetivo de demonstrar essa afirmação precisaremos de alguns resultados da Topologia Combinatória e da Álgebra.
In this article we give various formulates for compute the number of all coninvolutions over the group of upper triangular matrix with entries into the ring of Gaussian integers module p and the ring of Quaternions integers module p, with p an odd prime number.
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