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In this book, we define the concept of a Q-Smarandache implicative ideal of a Q-Smarandache BH-algebra, a Q-Smarandache implicative ideal with respect to an element of a Q-Smarandache BH-algebra in ordinary. Also, we gave the Q-Smarandache ( positive implictive ideal, translation ideal, commutative ideal, sub-commutative ideal, p-ideal and q-ideal) of a Q-Smarandache BH-algebra firstly, then we restricted these notions on Q-Smarandache implicative BH-algebra and Q-Smarandache medial BH-algebra. We stated and proved some theorems which determine the relationships among these notions and other types of ideals of a Q-Smarandache BH-algebra in ordinary algebra and by some examples. We showed that these notions are different. Also, we considered some properties of a Q- Smarandache implicative ideal of a Q-Smarandache BH-algebra in ordinary algebra and link them with these new notions.

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Prof. Husein Hadi Abbass is the Dean Assistant post for Scientific Affairs in the Faculty of Computer Science and Mathematics university of Kufa - Iraq. He has 50 researches published in local and international magazines in the fields of algebra, applied mathematics, image processing, genetic algorithms, and fuzzy mathematics.

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