Browsing by Author "Jiang Luh, Member"

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    The Semiring of Multisets
    (2000-11-13) Poplin, Phillip; Robert Hartwig, Chair; Ernest Stitzinger, Member; Jiang Luh, Member; Mohan Putcha, Member
    The aim of this paper is to develop the basic theory of the Multiset Algebra. Multisets provide a connection between the eigenvalue/eigenvector equations for the Max-Plus and nonnegative real number systems. Multisets provide a moregeneral setting than either of these two algebras. Results about these two algebras will be developed from the theory of Multisets.
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    The Staircase Decomposition for Reductive Monoids
    (2002-04-29) Burns, Brenda D.; Mohan Putcha, Chair; Thomas Lada, Member; Ernest Stitzinger, Member; Jiang Luh, Member
    The purpose of the research has been to develop a decomposition for the J-classes of a reductive monoid. The reductive monoid M(K) isconsidered first. A J-class in M(K) consists ofelements of the same rank. Lower and upper staircase matricesare defined and used to decompose a matrix x of rank r into theproduct of a lower staircase matrix, a matrix with a rank rpermutation matrix in the upper left hand corner, and an upperstaircase matrix, each of which is of rank r. The choice ofpermutation matrix is shown to be unique. The primary submatrix of a matrixis defined. The unique permutation matrix from the decompositionabove is seen to be the unique permutation matrix from Bruhat's decomposition for the primary submatrix. All idempotent elementsand regular J-classes of the lower and upper staircasematrices are determined. A decomposition for the upper and lowerstaircase matrices is given as well.The above results are then generalized to an arbitrary reductivemonoid by first determining the analogue of the components forthe decomposition above. Then the decomposition above is shown tobe valid for each J-class of a reductive monoid. Theanalogues of the upper and lower staircase matrices are shown tobe semigroups and all idempotent elements and regularJ-classes are determined. A decomposition for eachof them is discussed.