[5] **viXra:1309.0120 [pdf]**
*submitted on 2013-09-17 10:05:33*

**Authors:** Arsham Borumand Saeid

**Comments:** 63 Pages.

There are three types of Smarandache Algebraic Structures:
1.A Smarandache Strong Structure on a set S means a structure on S that has a proper subset P with a stronger structure.
2.A Smarandache Weak Structure on a set S means a structure on S that has a proper subset P with a weaker structure.
3.A Smarandache Strong-Weak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure.
By proper subset of a set S, one understands a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any.
Having two structures {u} and {v} defined by the same operations, one says that structure {u} is stronger than structure {v}, i.e. {u} > {v}, if the operations of {u} satisfy more axioms than the operations of {v}.
Each one of the first two structure types is then generalized from a 2-level (the sets P ⊂ S and their corresponding strong structure {w1}>{w0}, respectively their weak structure {w1}<{w0}) to an n-level (the sets Pn-1 ⊂ Pn-2 ⊂ … ⊂ P2 ⊂ P1 ⊂ S and their corresponding strong structure {wn-1} > {wn-2} > … > {w2} > {w1} > {w0}, or respectively their weak structure {wn-1} < {wn-2} < … < {w2} < {w1} < {w0}). Similarly for the third structure type, whose generalization is a combination of the previous two structures at the n-level.
A Smarandache Weak BE-Algebra X is a BE-algebra in which there exists a proper subset Q such that 1 belongs to Q, |Q| ≥ 2, and Q is a CI-algebra.
And a Smarandache Strong CI-Algebra X is a CI-algebra X in which there exists a proper subset Q such that 1 belongs to Q, |Q| ≥ 2, and Q is a BE-algebra.
The book elaborates a recollection of the BE/CI-algebras, then introduces these last two particular structures and studies their properties.

**Category:** Algebra

[4] **viXra:1309.0107 [pdf]**
*submitted on 2013-09-17 09:01:32*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 260 Pages.

The authors have constructed subset semirings using rings of both finite and infinite order. Thus, using finite rings we construct infinite number of finite semirings, both commutative as well as non-commutative, which is the main advantage of using this algebraic structure. For finite distribute lattices alone contribute for finite semirings.

**Category:** Algebra

[3] **viXra:1309.0027 [pdf]**
*submitted on 2013-09-05 21:38:31*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 246 Pages.

When the subsets of a loop are taken they also form only a subset groupoid and not a subset loop.
Thus the concept of subset interval loop is not there, and they only form a subset interval groupoid.
Subset matrix interval groupoid S using the loops Ln(m) has no S-Cauchy elements.

**Category:** Algebra

[2] **viXra:1309.0026 [pdf]**
*submitted on 2013-09-05 21:41:06*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 207 Pages.

In this book the authors introduce the notion of subset non associative semirings. It is pertinent to keep on record that study of non associative semirings is meager and books on this topic are still rare. Some open problems are suggested in this book.

**Category:** Algebra

[1] **viXra:1309.0019 [pdf]**
*submitted on 2013-09-04 21:27:50*

**Authors:** Eckhard Hitzer, Roxana Bujack, Gerik Scheuermann

**Comments:** 5 Pages. Proc. of the Third SICE Symposium on Computational Intelligence, August 30, 2013, Osaka University, Osaka, pp. 91-95.

Exactly 125 years ago G. Peano introduced the modern concept of vectors
in his 1888 book "Geometric Calculus - According to the Ausdehnungslehre (Theory of
Extension) of H. Grassmann". Unknown to Peano, the young British mathematician
W. K. Clifford (1846-1879) in his 1878 work "Applications of Grassmann's Extensive
Algebra" had already 10 years earlier perfected Grassmann's algebra to the modern
concept of geometric algebras, including the measurement of lengths (areas and volumes)
and angles (between arbitrary subspaces). This leads currently to new ideal methods for
vector field computations in geometric algebra, of which several recent exemplary results
will be introduced.

**Category:** Algebra