Completely Regular (?)-classes And Affine Spanning In Algebraic Monoids

The structure problem in the theory of linear algebraic monoids is studied in this thesis. It splits into 2 subtopics: completely regular (?)-classes, affinely spanned algebraic monoids.Let K be an algebraically closed field. Given a completely regular (?)-class J of a linear algebraic monoid over K, we construct a linear algebraic monoid with kernel J, and thus characterize the structure of completely regular (?)-classes. On the other hand, we consider the Schwarz radical of linear algebraic semigroups which is defined in semigroup theory, and define the radical of completely regular(?)-classes. We give some new characterization of the complete regularity, regularity and solvability of irreducible linear algebraic monoids in terms of radical data.Suppose K is an algebraically closed field of characteristic zero. Let Mn(K)be the algebra of all square matrices of order n. An algebraic monoid in Mn (K)is called an affinely spanned algebraic monoid, if it is an affine subspace of the space Mn(K). Affinely spanned algebraic monoids form a basis class of linear algebraic monoids. We use the information of non-units part of affinely spanned algebraic monoids to give some characterization of the structure of their unit groups. More-over, we prove that the set of all quasi-stochastic matrices of order n is a regular affinely spanned algebraic monoid, and its structure is studied.