We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley’s structure theorem for algebraic groups.

In this paper we give a construction of free 2-crossed modules. By the use of a `step-by-step' method based on the work of Andr e, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of the notion of a free 2-crossed module of algebras.

We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups...

The Cherlin-Zil’ber Conjecture states that all simple groups of finite Morley rank are algebraic. We prove that any minimal counterexample to this conjecture has a unique conjugacy class of Carter subgroups, which are defined as being the definable connected nilpotent subgroups of finite index in their normalizers, and which are analogous to Cartan subgroups in algebraic

Let G be an algebraic group over a field k. We call g ∈ G(k) real if g is conjugate to g−1 in G(k). In this paper we study reality for groups of type G2 over fields of characteristic different from 2. Let G be such a group over k. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in G(k) is real if and only if it is a product of two involutions in ...

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Diffalg(X) the group of algebraic diffeomorphisms of X into itself. The group Diffalg(X) acts diagonally on X n, for any natural integer n. We show that this action is transitive, for all n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebra...

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in C are established in this paper. The finite determination result gives sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for realanalytic generic submanifolds under appropriate assumptio...

A brief survey of real algebraic structures on topological spaces is given.

We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” addressing unital algebraic structure rather than simply vector space structure . One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of d...

Using results of Trnková, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal F-coalgebra. Several equivalent characterizations of boundedness are provided.