(Non-)Koszulness of operads for $$n$$ n -ary algebras, galgalim and other curiosities

OPERADS FOR n - ARY ALGEBRAS – CALCULATIONS AND CONJECTURES

In [8] we studied Koszulity of a family tAssd of operads depending on a natural number n ∈ N and on the degree d ∈ Z of the generating operation. While we proved that, for n ≤ 7, the operad tAssd is Koszul if and only if d is even, and while it follows from [4] that tAssd is Koszul for d even and arbitrary n, the (non)Koszulity of tAssd for d odd and n ≥ 8 remains an open problem. In this note ...

(NON-)KOSZULITY OF OPERADS FOR n-ARY ALGEBRAS, COHOMOLOGY AND DEFORMATIONS

We investigate Koszulity of operads for various n-ary algebras. We then focus to algebras with one anti-associative operation. Since the corresponding operad is not Koszul, the deformation cohomology differs from the standard one. We describe the relevant part of the deformation cohomology for this type of algebras using the minimal model for the antiassociative operad. In the remaining section...

On $n$-ary Hypergroups and Fuzzy $n$-ary Homomorphism

NEW TYPES OF FUZZY n-ARY SUBHYPERGROUPS OF AN n-ARY HYPERGROUP

In this paper, the new notions of ``belongingness ($in_{gamma}$)"and ``quasi-coincidence ($q_delta$)"  of a fuzzy point with a fuzzyset are  introduced. By means of this new idea, the  concept of$(alpha,beta)$-fuzzy $n$-ary subhypergroup of an $n$-aryhypergroup is given, where $alpha,betain{in_{gamma}, q_{delta},in_{gamma}wedge q_{delta}, ivq}$,  andit is shown that, in 16 kinds of $(alpha,beta...

Varieties of Anticommutative n-ary Algebras

A fundamental problem in the theory of n-ary algebras is to determine the correct generalization of the Jacobi identity. This paper describes some computational results on this problem using representations of the symmetric group. It is well known that over a field of characteristic 0 any variety of n-ary algebras can be defined by multilinear identities. In the anticommutative case, it is show...