Linear operators that preserve pairs of matrices which satisfy extreme rank properties

Extreme Preservers of Term Rank Inequalities Over Nonbinary Boolean Semiring

“…Let S be any semiring. An operator T : M m,n (S) → M m,n (S) is called a (P, Q, B)-operator if there exist permutation matrices P and Q, and a matrix B ∈ M m,n (S) with no zero entries, such that In [4] linear preservers of extremal cases of classical matrix inequalities over fields were characterized. On the other hand, linear preservers for various rank functions over semirings have been the object of much study during the last 30 years, see for example [2]- [12].…”

“…The investigation of the corresponding problems over semirings for the column rank function was done in [3]. The complete classification of linear operators that preserve equality cases in matrix inequalities over fields was obtained in [4]. For details on linear operators preserving matrix invariants one can see [9] and [10].…”

INJECTIVE LINEAR MAPS ON τ∞(F) THAT PRESERVE THE ADDITIVITY OF RAN

“…If (1.1) holds, then we say that φ preserves the additivity of rank, or we say that φ preserves pairs x, y satisfying extreme rank properties (as in [4] and [16]). It was shown in [9] that the above condition is equivalent to preserving the substractivity of rank.…”

Extreme Preservers of Maximal Column Rank Inequalities of Matrix Multiplications Over Semirings

“…Recently, Beasley and his colleagues investigated rank inequalities of matrices over semirings ( [1]) and characterized the linear operators that preserve extreme set of matrix pairs for rank inequality cases ( [2] and [3]). This research extends the linear preserver problems to the set of matrix pairs from the set of single matrices.…”