Orthomodular posets of idempotents in finite rings of matrices (Q1907559)

The set \(\text{Idem }R\) (\(\text{Herm Idem }R\)) of all idempotent (and selfadjoint) elements of a ring \(R\) with identity (and involution) forms an orthomodular poset by putting \(a \leq b\) iff \(ab = ba = a\) and \(a':= 1 - a\) \((a, b \in R)\). Orthomodular posets of this kind are studied for the ring \(R = \text{Mat} (n \times n, S)\) of all \(n \times n\) matrices over the ring \(S\) (with transposition as involution) where \(S\) is a finite residue class ring over the integers or a Galois field. It is proved that for a Galois field \({\mathbf F},\text{ Herm Idem(Mat} (n \times n, {\mathbf F}))\) is isomorphic to the orthomodular poset of all splitting linear subspaces of the vector space \({\mathbf F}^n\) over \({\mathbf F}\). (A linear subspace \(U\) of \({\mathbf F}^n\) is called splitting if \(U + U^\perp = {\mathbf F}^n\)).