LINEARITY OF $n$-ARY ASSOCIATIVE ALGEBRAS

n-ary associative algebras, cohomology, free algebras and coalgebras

When n is odd, a cohomology of type Hochschild for n-ary partially associative algebras has been defined in Gnedbaye’s thesis. Unfortunately, the cohomology definition is not valid when n is even. This fact is found again in the computations of the n-ary partially associative free algebra. In this work, we define in a first time two approachs of an Hochschild cohomology for n-ary partially asso...

Classification of All Associative Mono-n-ary Algebras with 2 Elements

We consider algebras with a single n-ary operation and a certain type of associativity.We prove that up to isomorphism there are exactly 5 of these associative mono-n-ary algebras with 2 elements for even n ≥ 2 and 6 for odd n ≥ 3. These algebras are described explicitly. It is shown that a similar result is impossible for algebras with at least 4 elements. An application concerning the assignm...

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

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...

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...