| Popis: |
Let D be a division ring, n ≥ 3 an integer, and P n ( D ) the poset of all n × n idempotent matrices over D with the partial order defined by P ≤ Q if P Q = Q P = P . Let T ∈ M n ( D ) be an invertible matrix and σ : D → D an endomorphism ( τ : D → D an anti-endomorphism). For any P ∈ P n ( D ) we denote by P σ ( P τ ) the idempotent matrix obtained from P by applying σ (τ) entrywise. The map ϕ : P n ( D ) → P n ( D ) defined by ϕ ( P ) = T P σ T − 1 , P ∈ P n ( D ) ( ϕ ( P ) = T t ( P τ ) T − 1 , P ∈ P n ( D ) ) is an injective map preserving order in both directions. Every such map is called a standard map. It has been known before that if D is an EAS division ring, then every injective order preserving map ϕ : P n ( D ) → P n ( D ) is either standard or of a very special degenerate form. In this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the EAS assumption is indispensable. Then we define generalized standard maps and using them we describe the general form of injective order preservers on P n ( D ) for an arbitrary division ring D . Our proof is shorter than the original one for the special case of EAS division rings. Under somewhat stronger assumptions we get two characterizations of standard maps. All the results are optimal as shown by counterexamples. |
Full Text from ScienceDirect