We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed Z action as a factor of a subaction of a Z-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with Z. Let H be a finitely generated group and G = Z o H a semidirect product. We show that for any effectively closed H-dynamical system ...

First, let m and n be positive integers such that n is odd and gcd(m, n) = 1. Let G be the semidirect product of cyclic groups given by G= Z8m Z2n = 〈x, y : x8m = 1, y2n = 1, and yxy−1 = x4m+1〉. Then the number of hamilton paths in Cay(G : x, y) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order...

A common generalization of the author's embedding theorem concerning the £-unitary regular semigroups with regular band of idempotents, and Billhardt's and Ismaeel's embedding theorem on the inverse semigroups, the closure of whose set of idempotents is a Clifford semigroup, is presented. We prove that each orthodox semigroup with a regular band of idempotents, which is an extension of an ortho...

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the Ehrenfest group. The underlying P...

We use ordered categories to study semidirect decompositions of finite ordered semigroups. We obtain ordered analogues of the derived category theorem and of the delay theorem. Next we prove that the ordered analogues of semilattices and of J -trivial monoids constitute local varieties, and we derive some decomposition theorems from these results. In [10], Pin and Weil initiated a study of the ...

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space. 1. Almost direct products of free...

If a graded Lie algebra is the direct sum of two graded sub Lie algebras, its bracket can be written in a form that mimics a ”double sided semidirect product”. It is called the knit product of the two subalgebras then. The integrated version of this is called a knit product of groups — it coincides with the ZappaSzép product. The behavior of homomorphisms with respect to knit products is invest...