Involutions on Incidence Algebras of Finite Posets

Abstract

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.

$(k+1)$-potent Matrices in triangular matrix Groups and Incidence Algebras of Finite Posets

“…). In general, consider the poset of the figure (b) then the proof follow similarly to [12] and we conclude that…”

“…Then, of the equation (11) we have x = a i,i+m . • Whereas, if a ii = a i+m,i+m and a i,i+m satisfies (1), then for the equation (12), x = a i,i+m as well. Summing up A k+1 = A…”

“…In this section, we fix the notation and recall some definitions and basic facts that will be used throughout the article. The notations and observations in this article are the same that in [12].…”

“…Then, in this case, using the matrix representation apply in [12], we obtain that N umber of q1,q2,••• ,qs in:…”

Coninvolutions on Upper Triangular Matrix Group over the Ring of Gaussian Integers and Quaternions integers modulo $p$

“…Denote by T n (R) the n × n upper triangular matrix group with entries in R. A matrix A ∈ T n (R) is called an Involution iff A 2 = I n with I n the identity in T n (R). Involutions on T n (R) has been studied by several authors, see for instance [2,4] also over Incidence Algebras see [3]. On the other hand, Coninvolution matrices has been studied by [1] and [5].…”