The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi’s and Schmitt’s more general fo...

It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterised as those adjunctions that preserve the order e...

This paper explicitly constructs cofree coalgebras over operads in the category of DG-modules. It is shown that the existence of an operadaction on a coalgebra implies a “generalized coassociativity” that facilitates the construction. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). In the pointed irreducible case, this constructio...

Crop residues are a key bulk feedstock for supplying renewable carbon bioenergy production and the broader bioeconomy without compromising food security. However, it is frequently advised to harvest no more than half of this potential ensure preservation soil organic (SOC) stocks. In study, we challenge recommendation demonstrate that crop residue allowing maintain long-term SOC stocks spatiall...

This article is divided into two parts. In the first part we work over a field k and prove that Frobenius space associated to algebra generated as left A-module by coproduct. particular, dimension coincides with of algebra. second commutative ring We introduce concept nearly algebras in this context construct solutions quantum Yang-Baxter equation starting from elements space. Also, give an alt...