Dirac reduction for nonholonomic mechanical systems and semidirect products

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange–Dirac and Hamilton–Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler–Poincaré–Suslov equations with advected parameters and the implicit Lie–Poisson–Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin–Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.     

Large subdirect product of modules as direct summand of their direct product

Abstract

The Zappa-Szép product of a pair of groups generalizes the semidirect product in that neither factor is assumed to be normal in the result. We extend the applicability of the Zappa-Szép product to multiplicative structures more general than groups with emphasis on categories and monoids. We also explore the preservation of various properties of the multiplication under the Zappa-Szép product.     

Zappa–Szép products of bands and groups

Zappa–Szép products arise when an algebraic structure has the property that every element has a unique decomposition as a product of elements from two given substructures. They may also be constructed from actions of two structures on one another, satisfying axioms first formulated by G. Zappa, and have a natural interpretation within automata theory. We study Zappa–Szép products arising from actions of a group and a band, and study the structure of the semigroup that results. When the band is a semilattice, the Zappa–Szép product is orthodox and ℒ-unipotent. We relate the construction (via automata theory) to the λ-semidirect product of inverse semigroups devised by Billhardt.     

Stability of Jensen functional equation on semigroups

In this paper we introduce a Jensen type functional equation on semigroups and study the Hyers-Ulam stability of this equation. It is proved that every semigroup can be embedded into a semigroup in which the Jensen equation is stable.     

On Commutativity of Ideal Extensions

In this article, we examine commutativity of ideal extensions. We introduce methods of constructing such extensions. In particular, we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a field. This answers a question from [1 Andruszkiewicz, R. R., Puczy?owski, E. R. (1995). On commutative idempotent rings. Proc. Roy. Soc. Edinburgh Sect. A 125(2):341–349.[Crossref] , [Google Scholar]]. Moreover, we classify fields of characteristic zero which can be obtained as T/I for some T.     

On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups

Let H be a locally compact group and K be a locally compact abelian group. Also let G=H× _τ K denote the semidirect product group of H and K, respectively. Then the unitary representation (U,L ²(K)) on G defined by is called the quasi regular representation. The properties of this representation in the case K=(ℝ ⁿ ,+), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) , and (ii) the stabilizers H ^ω are compact for a.e. ; both are necessary for square integrability of U. Furthermore, we shall consider some sufficient conditions for the square integrability of U. Also, for the square integrability of subrepresentations of U, we will introduce a concrete form of the Duflo-Moore operator.      

Catenarian Numbers

An integer n is called catenarian if, whenever L/K is an n-dimensional field extension, all maximal chains of fields going from K to L have the same length. Catenarian field extensions and catenarian groups are defined analogously. If n is an even positive integer, 6n is non-catenarian. If n ≥ 3 is odd, there exist infinitely many odd primes p such that p ² n is non-catenarian. A finite-dimensional field extension is catenarian iff its maximal separable subextension is. If q < p are odd primes where q divides p ? 1 (resp., q divides p + 1), every (resp., not every) group of order p ² q is catenarian.     

Semidirect products OF fuzzy subgroups

The semidirect product of two groups is an important construction in group theory. In this paper we define the semidirect product of fuzzy subgroups and give conditions for it to be a fuzzy subgroup. This extends the work of H. Sherwood on direct products of fuzzy subgroups. We then give an example where a fuzzy subgroup of a semidirect product is a semidirect product of fuzzy subgroups.