On Some Zero-Filiform Algebras

One-generated nilpotent bicommutative algebras

“…The description of one-generated, or cyclic, groups is well known: there exists a unique one-generated group of order n, up to isomorphism. In the case of algebras, similar results have been obtained in varieties such as associative [16], noncommutative Jordan [27], Leibniz and Zinbiel [38], since it was proved that there is only one ndimensional one-generated nilpotent algebra from each of them. However, this circumstance does not hold for every variety of non-associative algebras.…”

Some Classes of Nilpotent Associative Algebras

“…All null-filiform associative algebras were described in [12,Proposition 5.3]. 12]). An arbitrary n-dimensional null-filiform associative algebra is isomorphic to the algebra: µ n 0 : e i e j = e i+j , 2 ≤ i + j ≤ n, where {e 1 , e 2 , .…”

One-generated nilpotent terminal algebras

“…Another interesting direction is a study of one-generated objects. The description of one-generated finite groups is well-known: there is only one one-generated group of order n. In the case of algebras, there are some similar results, such that the description of n-dimensional one-generated nilpotent associative [12], noncommutative Jordan [22], Leibniz and Zinbiel algebras [38]. It was proven that there is only one n-dimensional one-generated nilpotent algebra in these varieties.…”