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The subject material in the book is presented in Five Chapters. In the First Chapter, we study the relation between Smarandache semirings and many-valued logics, in particular Basic logic and Lukasiewicz logic. We will also definitions of Smarandache semirings, automata on BL - algebras and MV- algebra. In the Second Chapter, we study the action of Legendre´s polynomials on commutative finite rings R, semirings, positively ordered commutative semirings and some other semirings. The rings Fq , Z/nZ where n is an RSA integer and commutative sub rings o f Matn (Fq) are the main examples. We study the difficulty of discrete Legendre´s problem in these rings and prove several equivalence results. In the Third Chapter, we deal, with Smaandache loops, sub loops, Smarandache sub loops, normal loops, Smarandache normal loops, the structure of Moufang loops and Paige. We analyze DLP (Discrete Logirithm Problem) in the Paige loop M (q), semirings positively ordered semirings, partially order semirings and some other semirings problems as based on exponentiation and conjugation in the Maugfang loop M (q). In the Fourth Chapter, we deal with an infinite family of finite semirings, Smara

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Dr.V. VasuDepartment of MathematicsS.V.University,Tirupati. Working as Teaching Assistant in Department of Mathematics, P.G. & Engineering, since 2003 is published 20 National and International papers

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