The staircase decomposition for reductive monoids

The purpose of the research has been to develop a decomposition for the $J$-classes of a reductive monoid. The reductive monoid M _n(K) is considered first. A $J$-class in M_n(K) consists of elements of the same rank. Lower and upper staircase matrices are defined and used to decompose a matrix x of rank r into the product of a lower staircase matrix, a matrix with a rank r permutation matrix in the upper left hand corner, and an upper staircase matrix, each of which is of rank r. The choice of permutation matrix is shown to be unique. The primary submatrix of a matrix is defined. The unique permutation matrix from the decomposition above is seen to be the unique permutation matrix from Bruhat's decomposition for the primary submatrix. All idempotent elements and regular $J$-classes of the lower and upper staircase matrices are determined. A decomposition for the upper and lower staircase matrices is given as well.; The above results are then generalized to an arbitrary reductive monoid by first determining the analogue of the components for the decomposition above. Then the decomposition above is shown to be valid for each $J$-class of a reductive monoid. The analogues of the upper and lower staircase matrices are shown to be semigroups and all idempotent elements and regular $J$-classes are determined. A decomposition for each of them is discussed.