A canonical representation of maximal, indecomposable semi groups of constant-rank nonnegative r-potent matrices in ݑ0ݑ(R)

Abstract

An ݑŸ-potent matrix in ݑ0ݑ›(R) is an ݑ› × ݑ› matrix satisfying ݐذݑŸ = ݐ¸. A multiplicative semigroup S in ݑ0ݑ›(R) is said to be decomposable if there exists a special kind of common invariant subspace called standard invariant subspace for each ݐ´ ∈ S. A semi-group S of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) is known to be decomposable if rank (ݑ†) > ݑŸ − 1 for all ݑ† in S. Further, a semigroup S in ݑ0ݑ›(R) of nonnegative matrices will be called a full semigroup if S has no common zero row and no common zero column. We have studied the structure of maximal semi-groups of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) under the special condition of fullness. The objectives of this paper are twofold: (1) To find conditions under which semigroups of nonnegative r-potent matrices can be expressed as a direct sum of maximal rank-one indecomposable semigroups of r-potent matrices; and (2) To obtain a canonical representation of maximal indecomposable rank-one semigroups of r-potent matrices which in the light of the above result gives a complete characterization of such semigroups having constant rank.