In this note by considering a complete lattice L, we define thenotion of an L-Fuzzy hyperrelation on a given non-empty set X. Then wedefine the concepts of (POM)L-Fuzzy graph, hypergraph and subhypergroupand obtain some related results. In particular we construct the categories ofthe above mentioned notions, and give a (full and faithful) functor form thecategory of (POM)L-Fuzzy subhypergroups ...

We derive from the compatibility of associators with module harmonic coproduct, obtained in Part I series, inclusion torsor into that double shuffle relations, which completes one aims this series. define two stabilizer torsors using and algebra coproducts I. show can be described torsor, latter is contained torsor.

In this paper, we first prove that any atom of a permutation obtained by the super-shuffle product two permutations can only consist some complete atoms original permutations. Then, and cut-box coproduct on are compatible, which makes it bialgebra. As algebra is graded connected, Hopf algebra.

This presentation provides a brief overview of the production and use of 1,3-butadiene in the United States. Starting as a coproduct of ethylene, the 1,3-butadiene monomer is extracted and purified, then transferred to consumers. Major uses of 1,3-butadiene include the manufacture of styrene-butadiene rubber, polybutadiene rubber, and adiponitrile.

Increased demand for ethanol as a fuel additive has resulted in dramatic growth in ethanol production. Ethanol is produced from corn by either wet milling or dry-grind processing. In wet milling, the corn kernel is fractionated into different components, resulting in several coproducts. Wet-milling plants are capital intensive because of equipment requirements; they produce large volumes of eth...

As studies continue to examine new value added uses for ethanol coproducts, it is important to have means to easily determine the feasibility of the processing steps involved. Many industries widely use computer simulation programs for this purpose, and for planning the use of resources and equipment capacities, and to determine processing costs. The objective of this project was to determine t...

Many industries operate technologies in which multiple outputs (co-products) are jointly produced. In some settings (“vertical”) the co-products differ along a performance dimension and are substitutable. In other settings (“horizontal”) the co-products differ in their applications and are not substitutable. In both cases, three important attributes of a co-product technology are its processing...

The class of quantum affinizations (or quantum loop algebras, see [Dr2, CP3, GKV, VV2, Mi1, N1, Jin, H3]) includes quantum affine algebras and quantum toroidal algebras. In general they have no Hopf algebra structure, but have a “coproduct” (the Drinfeld coproduct) which does not produce tensor products of modules in the usual way because it is defined in a completion. In this paper we propose ...

In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semi-direct coproduct of Hopf algebras.

In κ-deformed relativistic framework we consider three different definitions of κ-deformed velocities and introduce corresponding addition laws. We show that one of the velocities has classical relativistic addition law. The relation of velocity formulae with the coproduct for fourmomenta and noncommutative space-time structure is exhibited.