Regular algebraic monoids

No abstract. October 2, 1998     

Rationally smooth algebraic monoids

A reductive monoid M is called rationally smooth if it has sufficiently mild singularities as a topological space. We characterize this class of monoids in combinatorial terms. We then use our results to calculate the Betti numbers of certain projective, rationally smooth group embeddings using the “monoid BB-decomposition”.     

On algebraic semigroups and monoids,II

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.     

The cross section lattices and Renner monoids of the odd special orthogonal algebraic monoids

Let MSO _n (n is odd) be the special orthogonal algebraic monoid and M _n the monoid of all n × n matrices over an algebraically closed field. We will explicitly determine the cross section lattices Λ and the Renner monoids R of MSO _n by using admissible subsets (see Definition 3.1) and the Weyl group. It turns out that Λ is a sublattice of the cross section lattice of M _n and that R is a submonoid of the Renner monoid M _n . Also, we obtain some interesting properties of the submonoid (MSO _n ) _e = {y ∈ MSO _n | ye = ey = e} of MSO _n where e is an idempotent in MSO _n .     

Idempotent lattices, Renner monoids and cross section lattices of the special orthogonal algebraic monoids

Let MSO_n (n is even) be the special orthogonal algebraic monoid, T a maximal torus of the unit group, and the Zariski closure of T in the whole matrix monoid M_n. In this paper we explicitly determine the idempotent lattice , the Renner monoid , and the cross section lattice Λ of MSO_n in terms of the Weyl group and the concept of admissible sets (see Definition 3.1). It turns out that there is a one-to-one relationship between and the admissible subsets, and that is a submonoid of  , the Renner monoid M_n. Also Λ is a sublattice of Λ_n, the cross section lattice of M_n.     

Quasi-affine symmetric designs

A symmetric design with parameters v = q ²(q + 2), k = q(q + 1), λ = q, q ≥ 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P ₀, P ₁,..., P _q , P _q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ≥ 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2. Work of Sanjeevani Gharge is supported by Faculty Improvement Programme of U.G.C., India.     

Endomorphism monoids of acts over monoids

In this paper we investigate under which conditions a monoid R is defined by the endomorphism monoid of an act over R. More precisely, we ask when an isomorphism between two such endomorphism monoids over monoids R₁ and R₂ is induced by a semilinear isomorphism. The question is considered also for ordered and for topological monoids. On the way we characterize monoids over which all projective acts are free. An abstract of this paper appeared in the Proceedings of the Conference on Semigroups, Szeged 1972.     

Graph monoids

A graph semigroup refers to a monoid whose defining relations are of the form x_ix_j=x_jx_i. We describe the centralizer of an arbitrary element of a graph semigroup, show that there exists a unique factorization of any element into commuting parts, and prove related results. Dedicated to L. M. Gluskin on the occasion of his sixtieth birthday     

Finitary monoids

Abstract. No abstract.     

Tied monoids

We construct certain monoids, called tied monoids. These monoids result to be semidirect products finitely presented and commonly built from braid groups and their relatives acting on monoids of set partitions. The nature of our monoids indicate that they should give origin to new knot algebras; indeed, our tied monoids include the tied braid monoid and the tied singular braid monoid, which were used, respectively, to construct new polynomial invariants for classical links and singular links. Consequently, we provide a mechanism to attach an algebra to each tied monoid; this mechanism not only captures known generalizations of the bt-algebra, but also produces possible new knot algebras. To build the tied monoids it is necessary to have presentations of set partition monoids of types A, B and D, among others. For type A we use a presentation due to FitzGerald and for the other type it was necessary to built them.

     

Cohomology monoids of monoids with coefficients in semimodules II

We relate the old and new cohomology monoids of an arbitrary monoid M with coefficients in semimodules over M, introduced in the author’s previous papers, to monoid and group extensions. More precisely, the old and new second cohomology monoids describe Schreier extensions of semimodules by monoids, and the new third cohomology monoid is related to a certain group extension problem.