Each restriction semigroup is proved to be embeddable in a factorisable restriction monoid, or, equivalently, in an almost factorisable restriction semigroup. It is also established that each restriction semigroup has a proper cover which is embeddable in a semidirect product of a semilattice by a group.

We prove that the free Burnside Group B(3; 2) has order 27 and is isomorphic to ha; b j a3; b3(ab)3; (b 1a)3i. The technique of our proof is also used to show that ha; bja3; b3; a2(ba)nb2i is the semidirect product Cn2+n+1o C3.

We study a BGG-type category of infinite-dimensional representations of H[W ], a semidirect product of the quantum torus with parameter q, built on the root lattice of a semisimple group G, and the Weyl group of G. Irreducible objects of our category turn out to be parametrized by semistable G-bundles on the elliptic curve C∗/qZ .

The semidirect product of pseudovarieties of semigroups with an order-computable pseudovariety is investigated. The essential tool is the natural representation of the corresponding relatively free profinite semigroups and how it transforms implicit signatures. Several results concerning the behavior of the operation with respect to various kinds of tameness properties are obtained as applicati...

Motivated by the search for new examples of “noncommutative manifolds”, we study the noncommutative geometry of the group C*-algebras of various discrete groups. The examples we consider are the infinite dihedral group Z×σZ2 and the semidirect product group Z×σZ. We present a unified treatment of the K-homology and cyclic cohomology of these algebras.

The problem of decidability of membership in the semidirect product Sl L of the pseudovarieties Sl of all nite semilattices and L of all nite L-trivial semigroups has been around since 1984. This paper discusses several developments motivated by this question and leading to a proof that settles it aarmatively.

Following a general method proposed earlier, we construct here Wigner functions defined on coadjoint orbits of a class of semidirect product groups. The groups in question are such that their unitary duals consist purely of representations from the discrete series and each unitary irreducible representation is associated with a coadjoint orbit. The set of all coadjoint orbits (hence UIRs) is fi...

In this survey, we describe a general key exchange protocol based on semidirect product of (semi)groups (more specifically, on extensions of (semi)groups by automorphisms), and then focus on practical instances of this general idea. This protocol can be based on any group or semigroup, in particular on any non-commutative group. One of its special cases is the standard Diffie-Hellman protocol, ...