For efficiency evaluation of some of the Decision Making Units that have uncertain information, ROUGH Data Envelopment Analysis technique is used, which is derived from ROUGH set theorem and Data Envelopment Analysis (DEA). In some situations ROUGH data alter nonradially. To this end, this paper proposes additive ROUGH–DEA model and illustrates the proposed model by a numerical example.

In this paper we discuss some basic properties of ROUGH soft sets and hence we introduce the notion of ROUGH soft group. Basic properties of ROUGH soft group are presented and supported by some illustrative examples.

Let A be a Banach algebra and E be a Banach A -biMODULE then S=A ÅE, the l1-direct sum of A and E becomes a MODULE extension Banach algebra when equipped with the algebras product (a, x): (a', x')=(aa', a.x'+x.a'). In this paper, we investigate D-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the MODULE extension Banach algebra S=l1 (ES) Å l1 (S) is D-amenable as a l1 (ES) -MODULE if and only if l1 (ES) is amenable as Banach algebra.

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.

In this paper we define a congruence ~ on inverse semigroup S such that amenability of S is equivalent to amenability of S/ ~. We study MODULE amenability of semigroup algebra i1(S/ ~) when S is an inverse semigroup with idempotents E and prove that it is equivalent to MODULE amenability of i1 (S). The main difference of this action with the more studied trivial action is that in this case the corresponding homomorphic image is a Clifford semigroup rather than a discrete group.

The pattern of embedding of ROUGH approximation lattices defined by a reflexive relation in its Dedekind-MacNeille completion of ROUGH sets system is taken up for study in this work.The reflexive relation R for which the Dedekind-MacNeille completion of ROUGH sets system defined by R is the pasting of its ROUGH approximation lattices is characterized. Some properties of theDedekind-MacNeille completion of ROUGH sets system defined by areflexive relation R are also discussed.

In this paper, by considering the notion of extended BCK- MODULE, we define the concepts of free extended BCK-MODULE, free object in category of extended BCK-MODULEs and we state and prove some related results. Specially, we define the notion of idempotent extended BCK-MODULE and we get some important results in free extended BCK- MODULEs. In particular, in category of idempotent extended BCK-MODULEs, we give a method to make a free object on a nonempty set and in BCK- algebra of order 2, we give a method to make a basis for unitary extended BCK-MODULEs. Finally, we define the notions of projective and productive MODULEs and we investigate the relation between free MODULEs and projective MODULEs. In special case, we state the relation between free MODULEs and productive MODULEs.