We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM▹◂k\left(G\right)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_{𝕆}$ as transversal in an order 128 group $X$ with subgroup ${\mathbb{Z}}_2^3$ and hence obtain a Hopf quasigroup $k{G}_{𝕆}>◂k\left({\mathbb{Z}}_2^3\right)$ as a particular case of our construction.

On définit une structure de bigèbre différentielle graduée sur la somme directe des complexes cellulaires des permutoèdres, qui contient une sous-bigèbre différentielle graduée dont le complexe sous-jacent est la somme directe des complexes cellulaires des polytopes de Stasheff. Ceci étend des constructions de Malvenuto et Reutenauer et de Loday et Ronco pour les sommets des mêmes polytopes.

Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor ${ℍ}_C:C-Comod\to H_C-Comod$ that restricts to a representation equivalence ${ℍ}_C:C-comod\to H_C-como{d}_{sp}^{•}$, where $H_C$ is a coradical square complete hereditary bipartite...

Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor ${Hom}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-plethory is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A\left[X\right]$ is an initial object in the category of such structures. The $D$-algebra $Int\left(D\right)$ has such a structure if $D=A$ is a domain such that the natural $D$-algebra homomorphism ${\theta }_n:{{⨂}_D}_{i=1}^{n}Int\left(D\right)\to Int\left(D^n\right)$ is an isomorphism for...