Matrices as graded BiHom-algebras and decompositions

We present matrices as graded regular BiHom-associative algebras and study several features of their decompositions. More precisely, we introduce a notion of connection in the support of the grading and use it to construct a family of canonical graded ideals. We show that, under suitable assumptions, including Σ-multiplicativity, maximal length and triviality of the centre, the matrix regular BiHom-associative algebra decomposes into a direct sum of graded simple ideals. We further extend our results to general graded regular BiHom-associative algebras over arbitrary base fields. As applications, we reinterpret classical gradings on matrix algebras such as those induced by Pauli matrices and the ℤn × ℤn-grading in terms of our setting.