In this article we study two different generalizations of von Neumann REGULARITY, namely strong topological REGULARITY and weak REGULARITY, in the Banach algebra context. We show that both are hereditary properties and under certain assumptions, weak REGULARITY implies strong topological REGULARITY. Then we consider strong topological REGULARITY of certain concrete algebras. Moreover we obtain the following non-commutative analog of a result of Kaplansky. A bounded operator T on a Banach space X whose point spectrum sp(T) contains a nonzero complex number, is weakly regular.

The word "History" is used in the Persian Language in two meanings: 1. A collection of written scientific, theoretical and intellectual activities of the historians which is the same as "Science of History"; 2. A collection of incidents and events occurred by the human which implies history in its conventional meaning and is used without adding the word "Science". If we could determine the manner of recognition or acquiring knowledge of history in the second meaning, namely, as a collection of events, the existence of something called "Science of History" would also be proven, thus, we may say that the science of history is really a "Science". The present article by considering the ideas of scientists and historians from the earliest time up to the beginning of the twentieth century, distinguishes the two different meanings of history and follows the developments in the current outlooks about history as well as its REGULARITY and eventuality.

In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in residuated lattice a new type of filter called special filter and investigate its properties. Finally, regular filter property in residuated lattices is introduced and is studied in details.

In this article we give a characterization of monoids for which torsion freeness, ((principal) weak, strong) flatness, equalizer flatness or Condition (E) of finitely generated and (mono) cyclic acts and Condition (P) of finitely generated and cyclic acts implies REGULARITY. A characterization of monoids for which all (finitely generated, (mono) cyclic acts are regular will be given too. We also show that monoids for which all regular right acts are WPF, WKF, PWKF, TKF or satisfy Conditions (P), (WP) and (PWP) are the same as those for which all regular right acts are projective or strongly flat. Monoids S with E(S)ÍC(S) or those for which every element of E(S)\{1} is right zero will be characterized, when all (finitely generated, cyclic) right S-acts satisfying Conditions (PWP), (WP), (PE) or (P) are regular. Simple monoids for which all (finitely generated, cyclic) right acts with property U (U is a property of acts over monoids implied by Condition (P)) are regular will be characterized too.

In this paper, we giving certain properties of the commuting regular semigroups, we get significant results on semigroups. Our investigation involve certain interesting class of commuting regular semigroups.

In this paper, we study the Arens REGULARITY properties of module actions and we extend some proposition from Baker, Dales and Lau into general situations. We establish some relationships between topological centers of module actions and factorization properties of them with some results in group algebras.

Let A be a Banach algebra, A’ ’ a Banach A-module. In this paper, we give a simple criterion for the Arens REGULARITY of a bilinear mapping on normed spaces, which applies in particular to Banach module actions, and them investigate those conditions under which the second adjoint of a derivation into a dual Banach algebra module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included. A^(4) is a banach algebra with four Arens products. The bilinear map T is Arens regular when the equality T*** = T^( r***r ). If T: A × A’ ’ → A’ ’ is multiplication left module on A, the following statements are equivalent, i: T is regular ii: T**** = T^(r****r) iii: T****( A’ ’ ’ , A’ ’ ) ⊆ A’ ’ ’ iv: the linear map a → T*( a’ ’ ’ , a): A → A’ ’ ’ is weakly compact for every a’ ’ ’ ∈ A’ ’ ’ . Also If module actions are regular, then every inner derivation D: A → A’ ’ ’ is weakly compact; moreover, D**: (A’ ’ , □ ) → A^(5) and D**: (A’ ’ , ⋄ ) → A^(5) are also inner derivation.

We present a characterization of Arens regular semi-group algebras L1 (S), for a large class of semigroups. Mainly, we show that if the set of idempotents of an inverse semigroupS isfinite, then L1 (S) is Arens regular if and only if S is finite.