In 2014, Vargas first defined a super-shuffle product and cut-box coproduct on permutations. 2020, Aval, Bergeron Machacek introduced the labeled simple graphs. this paper, we generalize from graphs to (0,1)-matrices. Then prove that vector space spanned by (0,1)-matrices with is graded algebra coalgebra.

Wet coproducts fed to beef cattle include processing coproducts of the fruit, vegetable, juice, and brewing industries. Considerations for their utilization in beef cattle diets include quantity available, feeding value, quality of animal products produced, economics (e.g., transportation of water), storage and preservation, consumer perception, nuisance concerns, contaminants, and interactions...

1. Introduction. In the present note we announce some results which have been obtained by the systematic exploitation of filtration arguments in the study of (infinite dimensional) Lie algebras. The fundamental theorem along this line appears in the following section. In §3 we indicate how this result can be used to provide a new proof and at the same time a generalization of the Shirshov-Witt ...

We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than com...

Abstract. Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties. Characters of covariant tensor irreducible representations of the classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur fun...

The coproduct of a Feynman diagram is set up through identifying the perturbative unitarity of the S-matrix with the cutting equation from the cutting rules. On the one hand, it includes all partitions of the vertex set of the Feynman diagram and leads to the circling rules for the largest time equation. Its antipode is the conjugation of the Feynman diagram. On the other hand, it is regarded a...

By means of a new notion subforests an angularly decorated rooted forest, we give combinatorial construction coproduct on the free Rota–Baxter algebra forests. We show that this equips with bialgebra structure and further Hopf structure.