Let L be a Lie CROSSED MODULE and Actpi(L) and Actz(L) be the pointwise inner actor and center actor of L, respectively. We will give a necessary and sufficient condition under which Actpi(L) and Actz(L) are equal.

Using free simplicial algebras with given CW−basis, it is shown how to construct a free or totally free quasi 3-CROSSED MODULE on suitable construction data. Quasi 3-CROSSED complexes are introduced and similar freeness results are given for these are discussed.

In the definition of a CROSSED MODULE (T, G,  ), the actions of the group T and G on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary CROSSED MODULE. Therefore, we get the category GCM, of all generalized CROSSED MODULEs and generalized CROSSED MODULE morphisms between them, and investigate some of its categorical properties. In particular, we study the relations between epimorphisms and the surjective morphisms, and thus generalize the corresponding results of the category of (ordinary) CROSSED MODULEs. By generalizing the conjugation action, we can find out what is the superiority of the conjugation to other actions. Also, we can find out that a generalized CROSSED MODULE with which other actions (other than the conjugation) has the properties similar to a CROSSED MODULE.

In this paper, we present algebroids and CROSSED MODULEs of algebroids. We also define pullback CROSSED MODULE of algebroids.

In this work we deal with actions of vector groupoid which is a new concept in the literature. After we give the definition of the action of a vector groupoid on a vector space, we obtain some results related to actions of vector groupoids. We also apply some characterizations of the category and groupoid theory to vector groupoids. As the second part of the work, we define the notion of a CROSSED MODULE over a vector groupoid. Finally, we show that the category Vg of the vector groupoids is equivalent to the category CModVg of the CROSSED MODULEs over a vector groupoid.

In this paper, we presented algebroids and CROSSED MODULEs of algebroids. We also presented pullback CROSSED MODULE of algebroids and we then define pushout CROSSED MODULEs of algebroids.

The theory of polygroups is a natural extension of group theory, where the composition of two elements results in a set rather than an element. This concept has found applications in diverse fields such as geometry, lattices, combinatorics, and color schemes. Addition-ally, the study of CROSSED MODULEs and their applications has played a crucial role in category theory, homology and cohomology of groups, homotopy theory, algebra, k-theory, and more. This paper presents the definition of a polyfunctor and transformation for polygroups, as well as the introduction of the concept of symmetric CROSSED MODULE to sym-metric CROSSED polyMODULEs. Our findings extend the classical results of CROSSED MODULEs to CROSSED polyMODULEs of polygroups.

The aim of this paper is to introduce the notion of cat1−, groupoids which are the groupoid version of cat1−, groups and to prove the categorical equivalence between CROSSED MODULEs over groupoids and cat1−, groupoids. In section 4 we introduce the notions of CROSSED squares over groupoids and of cat2−, groupoids, and then we show their categories are equivalent. These equivalences enable us to obtain more examples of groupoids.

In this paper we defined the concept of MODULE amenability of Banach algebras and MODULE connes amenability of MODULE dual Banach algebras. Also we assert the concept of MODULE Arens regularity that is different with [1] and investigate the relation between MODULE amenability of Banach algebras and connes MODULE amenability of MODULE second dual Banach algebras. In the following we study the relation between MODULE amenability, weak MODULE amenability and MODULE approximate amenability of Banach algebra. The notation of amenability of Banach algebras was introduced by B. Johnson in [7]. A Banach algebra A is amenable if every bounded derivation from A into any dual Banach A-biMODULE is inner, equivalently if H(A; X) = 0 for any Banach A-biMODULE X, where H(A; X) is the first Hochschild co-homology group of A with coefficient in X. Also, a Banach algebra A is weakly amenable if H(A; A) = 0. Bade, Curtis and Dales introduced the notion of weak amenability on Banach algebras in [4]. They considered this concept only for commutative Banach algebras. After that Johnson defined the weak amenability for arbitrary Banach algebras.