In this paper, we define an Fs-binary relation on a pair of Fs-sets and also define a partial order on the collection of all Fs-binary relations between the given pair of Fs-sets and prove that the collection with this given partial order is an infinitely distributive lattice. An Fs-set is a four triple in which first two components are crips sets such that the second component is sub set of first component. The fourth component is a complete Boolean algebra which is also the co-domain of two sub component functions while the third component is a function with combinations of two sub components given in which the first sub component is a complete Boolean valued function with first component as its domain and second sub component is another complete Boolean valued function with the second component as its domain and both sub component functions have fourth component as theirs co-domain and also the first sub component function is more valued than the second sub component function value.The third component which is the combination of two sub components is called the membership function of the given Fs-set. Here, the so called sub components are given within simple brackets after the third component.
In this paper, based upon Fs-set theory [Yogesara V, Srinivas G, Rath B. A theory of Fs-sets, Fs-complements and Fs-de Morgan laws. IJARCS. 2013;4(10)], we define Fs-Cartesian product of given family Fs-subsets of give Fs-set and we prove Axiom of choice for Fs-sets and we study the validity of converse of the Axiom of choice for Fs-sets.
In this paper, based upon Fs-set theory [1], we define a crisp Fs-points set FSP( ) for given Fs-set and establish a pair of relations between collection of all Fs-subsets of a given Fs-set and collection of all crisp subsets of Fs-points set FSP( ) of the same Fs-set and prove one of the relations is a meet complete homomorphism and the other is a join complete homomorphism and search properties of relations between Fs-complemented sets and complemented constructed crisp sets via these homomorphisms.
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