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# DSm Super Vector Space of Refined Labels : Volume 2

## By Kandasamy, W. B. Vasantha

Book Id: WPLBN0002828222
Format Type: PDF eBook:
File Size: 7.75 MB
Reproduction Date: 7/17/2013

 Title: DSm Super Vector Space of Refined Labels : Volume 2 Author: Kandasamy, W. B. Vasantha Volume: 2 Language: English Subject: Collections: Historic Publication Date: 2013 Publisher: World Public Library Member Page: Florentin Smarandache Citation APA MLA Chicago B. Vasantha Kandasam, B. W., & Smarandache, F. (2013). DSm Super Vector Space of Refined Labels : Volume 2. Retrieved from http://self.gutenberg.org/

Description
In this book authors for the first time introduce the notion of supermatrices of refined labels. Authors prove super row matrix of refined labels form a group under addition. However super row matrix of refined labels do not form a group under product; it only forms a semigroup under multiplication. In this book super column matrix of refined labels and m Å~ n matrix of refined labels are introduced and studied. We mainly study this to introduce to super vector space of refined labels using matrices.

Summary
We in this book introduce the notion of semifield of refined labels using which we define for the first time the notion of supersemivector spaces of refined labels. Several interesting properties in this direction are defined and derived.

Excerpt
THEOREM 1.1.1: Let S = {(a1 a2 a3 | a4 a5 | a6 a7 a8 a9 | … | an-1, an) | ai ∈ R; 1 ≤ i ≤ n} be the collection of all super row vectors with same type of partition, S is a group under addition. Infact S is an abelian group of infinite order under addition. The proof is direct and hence left as an exercise to the reader. If the field of reals R in Theorem 1.1.1 is replaced by Q the field of rationals or Z the integers or by the modulo integers Zn, n < ∞ still the conclusion of the theorem 1.1.1 is true. Further the same conclusion holds good if the partitions are changed. S contains only same type of partition. However in case of Zn, S becomes a finite commutative group.