Let X = (X, H) and Y = (Y, K) be two association schemes. In [1], an (external) semidirect product Z = (X × Y, H nπ K) of Y and X relative to π is defined. In this paper, given a normal closed subset N of K which is fixed by π, we find a way to construct a fusion scheme of Z with respect to N . We then show that every association scheme obtained by any of three products, direct, wreath and semi...

Fountain and Gomes have shown that any proper left ample semigroup embeds into a so-called W -product, which is a subsemigroup of a reverse semidirect product T ⋉ Y of a semilattice Y by a monoid T , where the action of T on Y is injective with images of the action being order ideals of Y. Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim o...

This paper shows how to reduce a Hamiltonian system on the cotangent bundle of a Lie group to a Hamiltonian system in the dual of the Lie algebra of a semidirect product. The procedure simplifies. unifies, and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden, Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids, magnetohydrodynamics, elasticity, the Max...

For matroids M and N on disjoint sets S and T , a semidirect sum of M and N is any matroid K on S ∪ T that, like the direct sum and the free product, has the restrictionK|S equal toM and the contractionK/S equal toN . We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension ofM on the set S ∪ T with the direct sum ofN and the ra...

In this paper we consider  (extended) metaplectic representation of the  semidirect product  $G_{mathbb{J}}=mathbb{R}^{2d}timesmathbb{J}$  where $mathbb{J}$ is a closed subgroup of $Sp(d,mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Fina...

It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.

Within the usual semidirect product S ∗ T of regular semigroups S and T lies the set Reg (S ∗ T ) of its regular elements. Whenever S or T is completely simple, Reg (S ∗T ) is a (regular) subsemigroup. It is this ‘product’ that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, U and V, the e-variety U...

It is proved that the automorphism group of a semigroup being an inflation of its proper subsemigroup decomposes into a semidirect product of two groups one of which is a direct sum of full symmetric groups.