The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of (α s , β r )derivation, (α s , β r )-quasiderivation and generalized (α s , β r )-derivation of 3-BiHom-Lie superalgebras, and their relation with derivation of BiHom-Lie superalgebras. We introduce also the concepts of Rota-Baxter operators and Nijenhuis Operators of BiHom 3-Lie superalgebras. We also explore the construction of 3-BiHom-Lie superalgebras by using Rota-Baxter of BiHom-Lie superalgebras. also been studied in connection with many areas of mathematics and physics, including combinatorics, number theory, operators and quantum field theory [16]. Furthermore, Rota-Baxter operators on a Lie algebra are an operator form of the classical Yang-Baxter equations and contribute to the study of integrable systems [5]. Further Rota-Baxter 3-Lie algebras are closely related to pre-Lie algebras [6]. Rota-Baxter of multiplicative 3-ary Hom-Nambu-Lie algebras were introduced by Sun and Chen, in [14].Deformations of n-Lie algebras have been studied from several aspects. See [1,9] for more details. In particular, a notion of a Nijenhuis operator on a 3-Lie algebra was introduced in [15] in the study of the 1-order deformations of a 3-Lie algebra. But there are some quite strong conditions in this definition of a Nijenhuis operator. In the case of Lie algebras, one could obtain fruitful results by considering one-parameter infinitesimal deformations, i.e. 1-order deformations. However, for n-Lie algebras, we believe that one should consider (n − 1)-order deformations to obtain similar results. In [9], for 3-Lie algebras, the author had already considered 2-order deformations. For the case of Hom-Lie superalgebras, the authors in [13] give the notion of Hom-Nijenhuis operator.Thus it is time to study 3-BiHom-Lie superalgebras, Rota-Baxter algebras and Nijenhuis operator together to get a suitable definition of Rota-Baxter of 3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras. Similarly, we give the relationship between Nijenhuis operator of 3-BiHom-Lie superalgebras and BiHom-Lie superalgebras. This paper is organized as follows: In Section 1, we recall the concepts of BiHom-Lie superalgebras and introduce the notion of 3-BiHom-Lie superalgebras. The construction of 3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras are established in Section 2. In section 3, we give the definition of (α s , β r )-derivation and (α s , β r )quasiderivation of 3-BiHom-Lie superalgebras. In section 4, we give the definition of Rota-Baxter of 3-BiHom-Lie superalgebras and the realizations of Rota-Baxter of 3-BiHom-Lie superalgebras from Rota-Baxter BiHom-Lie superalgebras. The Section 5 is dedicated to study the second order deformation of 3-BiHom-Lie superalgebras, and introduce the notion of Nijenhuis operator on 3-BiHom-Lie superalgebras, which could generate a trivial deformation. In the other part of this section we give some properties and results of Nijen...

We show that given a hom-Lie algebra one can construct the n-ary hom-Lie bracket by means of an (n − 2)-cochain of the given hom-Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov-Jacobi identity, thereby inducing the structure of n-hom-Lie algebra. We introduce the notion of a hom-Lie n-tuple system which is the generalization of a hom-Lie triple system. We construct hom-Lie n-tuple system using a hom-Lie algebra.Mathematics Subject Classification. 17A30, 17A36, 17A40, 17A42.In [6], generalizations of n-ary algebras of Lie type and associative type by twisting the identities using linear maps were introduced. The notions of representations, derivations, cohomology and deformations were studied in [3,12,15,21,24]. These generalizations include n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-Lie algebras (called also n-ary Nambu-Lie algebras) and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. In [4], a method was demonstrated how to construct ternary multiplications from the binary multiplication of a hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions; and it was shown that this method can be used to construct ternary hom-Nambu-Lie algebras from hom-Lie algebras. This construction was generalized to n-Lie algebras and n-hom-Nambu-Lie algebras in [5].It is well known that algebras of derivations and generalized derivations are very important in the study of Lie algebras and its generalizations. The notion of δ-derivation appeared in the paper of Filippov [14]. The results for δ-derivations and generalized derivations were studied by many authors. For example, Zhang and Zhang [26] generalized the above results to the case of Lie superalgebras; Chen, Ma, Ni and Zhou considered the generalized derivations of color Lie algebras, hom-Lie superalgebras and Lie triple systems [10,11]. Derivations and generalized derivations of n-ary algebras were considered in [17,18] and other papers. In [9], the authors generalize these results in the color n-ary hom-Nambu case.This paper is organized as follows. In Sect. 1, we review some basic concepts of hom-Lie algebras, n-ary hom-Nambu algebras and n-hom-Lie algebras. We also recall the definitions of derivations, α k -derivations, α kquasiderivations and α k -centroid. In Sect. 2, we provide a construction procedure of n-hom-Lie algebras starting from a binary bracket of a hom-Lie algebra and multilinear form satisfying certain conditions. To this end, we give the relation between α k -derivations, (resp. α k -quasiderivations and α kcentroid) of hom-Lie algebras and α k -derivations (resp. α k -quasiderivations and α k -centroid) of n-hom-Lie algebras. In Sect. 3, we introduce the notion of a hom-Lie n-tuple system which is the generalization of a Lie n-tuple system which is introduced in [13]. We construct a hom-Lie n-tuple system using a ho...

We introduce the concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via the cohomology theory.