Marczewski-Burstin Representation of Soft Algebras

Abstract

Using Marczewski and Burstin’s ideas, we present a general scheme for defining the concepts of (s)-sets and (s⁰)-sets in soft environments. We show that the classes of all soft (s)-sets and soft (s⁰)-sets are a soft algebra and a soft ideal, respectively. The soft (s)-sets and soft (s⁰)-sets are defined with respect to a family of non-null soft sets, which is known as a basis. A soft algebra is said to be a soft Marczewski-Burstin algebra if it is equal to the class of (s)-sets for some basis. The same is true for the respective soft ideal. Essential properties and characterizations of soft Marczewski-Burstin algebras are studied. As an application to the scheme, if the basis of the classes of soft (s)-sets and soft (s⁰)-sets is a soft topology, we show that soft Marczewski-Burstin algebra and soft Marczewski-Burstin ideal are respectively equal to the soft nowhere dense boundary algebra and the soft nowhere dense ideal. In this case, such a soft algebra and a soft ideal are called soft topological, which means each soft topological Marczewski-Burstin algebra is a soft Marczewski-Burstin algebra. The reverse is generally incorrect, which is verified by an example. In addition, some properties of soft topological Marczewski-Burstin algebras are also obtained.