Matrix Space To Keep The Rank Commutative Additive Mapping

Let n ≥ 2 be a given integer. Let k ≥ 2 be an arbitrary given integer. Let F be a field of characterstic 2 and M_n(F) be the set of all n x n matrices on F and T_n(F) be the set of all n x n upper triangular matrices on F. Suppose M is a matrix set over F and V is a subset of M × … × M .Suppose φ is a map from M into itself. Then φ is called a strong preserver k— tuple of matrices A₁, …, A_k rank permutable on M if φ satisfying rank = rank if and only if rank (A₁A₂ … A_k) = rank , for all σ∈ S_k, where (A₁, …, A_k) ∈ V, S_k is the symmetric group on k elements; φ iscalled a strong preserver k— tuple of matrices A₁, …, A_k rank reverse per-mutable on M if σ = ∈ S_k. For k = 2, φ are calledstrong preserver of rank commutativity on M.In Chapter 2 of this paper, we characterize the additive surjective map on M_n(F) that strong preserve the set of rank commutativity, and the additive surjective map on M_n(F) that strong preserve the set of rank reverse permutability matrix k—tuples and rank strong permutability matrix k—tuples. In Chapter 3, We characterize the additive surjective map on T_n(F) that strong preserve the set of rank commutativity, and the additive surjective map on T_n(F) that strong preserve the set of rank reverse permutability matrix k—tuples and rank strong permutability matrix k—tuples.