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We give a classification of noncommutative algebraic monoid structures on normal affine varieties such that the group of invertible elements of the monoid is connected, solvable, and has a one-dimensional unipotent radical. We describe the set of idempotents and the center of such a monoid and give a criterion for existence of the zero element. [ABSTRACT FROM AUTHOR]

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors. [ABSTRACT FROM AUTHOR]

We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n − 1 or 𝑛 on a normal affine variety of dimension 𝑛. [ABSTRACT FROM AUTHOR]

We study commutative associative polynomial operations 𝔸 n × 𝔸 n → 𝔸 n with unit on the affine space 𝔸 n over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties. [ABSTRACT FROM AUTHOR]

Consider an algebraic semigroup S and its closed subscheme of idempotents, E( S). When S is commutative, we show that E( S) is finite and reduced; if in addition S is irreducible, then E( S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E( S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E( S) is smooth, and its connected components are conjugacy classes of the unit group. [ABSTRACT FROM AUTHOR]

Abstract: An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid, G its unit group, P a parabolic subgroup of G, and a minimal idempotent. We show that and that the assignment defines a one-to-one correspondence between parabolic subgroups of G and of . [Copyright &y& Elsevier]