On exponent matrices of tiled orders
Mikhailo Dokuchaev, Vladimir Vasil'evich Kirichenko, Makar Plakhotnyk · Journal of Algebra and Its Applications · 2016
We describe two methods to determine all generators of the additive semigroup of the non-negative exponent [Formula: see text]-matrices, and illustrate them finding all generating [Formula: see text]-exponent matrices. The generating [Formula: see text]-exponent matrices are found using a computer. We consider the Hasse diagram [Formula: see text] of the partially ordered set of non-negative matrices and prove that for an arbitrary non-negative exponent matrix [Formula: see text] there exists an oriented path in [Formula: see text] starting in some matrix unit and ending in [Formula: see text] which does not pass through any other exponent matrix. We also show that for any non-negative exponent matrix [Formula: see text] there exists a chain of non-negative exponent matrices [Formula: see text] such that [Formula: see text] is a [Formula: see text]-matrix, and each [Formula: see text] is obtained from [Formula: see text] by adding a [Formula: see text]-matrix.