Decomposability and structure of nonnegative bands in \({\mathcal M}_n(\mathbb{R})\) (Q1300847)

A standard subspace of \(\mathbb{R}^n\) is a space spanned by a subset of the standard basis \(\{e_1,e_2,\dots,e_n\}\). A multiplicative semigroup \(S\) in \(M_n(\mathbb{R})\) is said to be decomposable if its members have a common nontrivial standard invariant subspace. Necessary and sufficient conditions for decomposability of nonnegative semigroups are given. In particular, decomposability of nonnegative bands (semigroups of idempotents) and their structure is discussed. It is proved that a nonnegative band with each member having rank greater than 1 is decomposable. Also, a geometric characterization of maximal, rank-one nonnegative bands is given.