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BiHom-groups are a nonassociative generalization of groups with the associativity law replaced by a twisted associativity law: α(a)(bc)=(ab)β(c), where α and β are two multiplicative mappings. The objective of this paper is to introduce and study BiHom-groups. We define the concept and provide examples of BiHom-groups. Additionally, we establish fundamental properties such as normal BiHom-subgroups, BiHom-quotient groups, and homomorphisms of BiHom-groups, and present the first, second, and third isomorphism theorems for BiHom-groups.

The main goal of this work is to introduce a notion of Hom-M-dendriform algebras which can be seen as the dendriform version of Hom-Malcev algebras. In fact they are the algebraic structures behind the O -operator of Hom-pre-Malcev algebras. They also fit into a larger framework as Hom-Malcev algebraic analogues of Hom-L-dendriform algebras. Furthermore, we show a connections between Hom-M-dendriform algebras and Hom-alternative quadri-algebras.

The purpose of this paper is to study pseudo-Euclidean and symplectic Hom-alternative superalgebras and discuss some of their proprieties and provide construction procedures. We also introduce the notion of Rota-Baxter operators of pseudo-Euclidean Hom-alternative superalgebras of any weight and Hom-post-alternative superalgebras. A Hom-post-alternative superalgebra consists of three operations such that some compatibility conditions are satisfied. We show that a weighted Rota-Baxter operator induces a Hom-post-alternative superalgebra naturally. Conversely, a Hom-post-alternative superalgebra gives rise to a new Hom-alternative superalgebra. In particular, a Hom-pre-alternative superalgebra is naturally built via symplectic structures.

We extend the concept of Rota-Baxter operator from algebras and groups to the setting of Hom-groups. The concept of Rota-Baxter Hom-group is introduced, and fundamental properties are described. Constructions of Rota-Baxter operators for Hom-groups are obtained. We obtain Rota-Baxter Hom-subgroups, quotient Rota-Baxter Hom-groups, and demonstrate the first, second, and third isomorphism theorems of Rota-Baxter Hom-groups. A theorem on extension of the Rota-Baxter operator of a Hom-group G to the Rota-Baxter operator of a Hom-group algebra over a field K, and the conditions of existence of a homomorphism from the Hom-algebra KG into itself, through a Rota-Baxter operator of Hom-group, are obtained.

The purpose of this paper is to introduce and give some constructions results and examples of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras. Also, we establish the bimodules and matched pairs of transposed Hom-Poisson algebras. Their related relevant properties are also given. Finally, the notion of O-operator is exploited to establish the relations between transposed Hom-Poisson and Hom-pre-Lie Poisson algebras. 

The aim of this paper is to give some constructions results of averaging operators on Hom-Lie algebras. The homogeneous averaging operators on q-deformed Witt and q-deformed W(2, 2) Hom-algebras are classified. As applications, the induced Hom-Leibniz algebra structures are obtained and their multiplicativity conditions are also given. 

In this paper, first we introduce the notion of crossed homomorphisms on Lie conformal algebras, then we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by crossed homomorphisms on Lie conformal algebras. This permits us to define cohomology for crossed homomorphisms. As an application, we study infinitesimal deformations, formal deformations and extendibility of finite order deformations of crossed homomorphism between Lie conformal algebras in terms of the cohomology theory.

In this paper, fixed point results with respect to generalized rational contractive mappings in semi-metric spaces endowed with a directed graph are proved. Some examples are provided to illustrate the results. The obtained results extend, improve and generalize many results in the existing literature.

The purpose of this article is to generalize (n − 1)-ary generalized derivations over n-hom-Lie algebras to the new class of (α, β, n)-hom-Lie algebras introduced by the authors. We study the action of the adjoint maps over these algebras, generalizing the concept of inner derivations. We examine the relation between the newlyfound bi-generalized derivation operators, adjoint maps and finite-dimensional extensions of a given (α, β, n)-hom-Lie algebra. We discuss multiplicativity in the algebra, the reduction of it by a bi-generalized derivation and extensions given by generalized trace operators.

This work focuses on the properties and structures of Hom-Lie algebras of generalized sl(2)-type. We construct classes of linear twisting maps that turn a skew-symmetric algebra of generalized sl(2)-type into a Hom-Lie algebra, and identify subclasses that result in multiplicative Hom-Lie algebras. We explore ideals, Hom-ideals, subalgebras, and Hom-subalgebras, emphasizing derived series, central descending series, and nilpotence and solvability properties. We also investigate the invariance of these subalgebras under the linear twisting maps and determine whether these subalgebras are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals. In particular, we examine these subalgebras and properties for the subfamilies of non-multiplicative Hom-Lie algebras of generalized sl(2)-type, highlighting the differences between non-multiplicative and multiplicative cases.