International Journal of Scientific and Research Publications, Volume 9, Issue 1, January 2019 788 ISSN 2250-3153 http://dx.doi.org/10.29322/IJSRP.9.01.2019.p8598 www.ijsrp.org Construction of B-Algebra on Cycles of Aunu Permutation Pattern * M. A. Mungadi 1 A.A. Haliru 1 S. Abdulrahman. 2 and M. S. Magami 3 1 Department of Mathematics Kebbi State University of Science and Technology, Aliero P.M.B. 1144 Aliero, Kebbi State. 2 Department of Mathematics Federal University Birnin Kebbi , Kebbi State. 3 Department of Mathematics,Usmanu Danfodiyo University,P.M.B. 2346 Sokoto, Nigeria. *Corresponding Author: maliyumunggadi19@gmail.com DOI: 10.29322/IJSRP.9.01.2019.p8598 http://dx.doi.org/10.29322/IJSRP.9.01.2019.p8598 Abstract Application of some algebraic properties of B-Algebra in AUNU Permutation Patterns was developed. Then the cycles AUNU Group were applied to form a table that satisfied some properties of B-algebra and some relevant theorem that will back our constructions was proposed. We also applied B-algebra in graph theory were we obtained a directed graph its adjency matrix Keywords: AUNU Groups, AUNU Patterns, B-Algebra, Permutations, AUNU Number and Graph. 1. Introduction AUNU permutation pattern are a class of (123) and (132)-avoiding permutation pattern with special properties associated with succession schemes. These patterns have been used extensively in formulation of different structures such as group, graph and in study of different algebra schemes such as cyclic designs and circuit design see Mustapha and Ibrahim, (2013); Ibrahim, (20005); Magami et al, (2012). Some other theoretic properties of the AUNU Pattern and AUNU group were identify and discussed especially in relation to integer modulo group see Usman and Ibrahim, (2011). Ibrahim and Saidu, (2016.) Present non- associative and non commutative properties of 123-avoiding pattern of AUNU permutation pattern. The research describe how non associative and non commutative properties can be established by using the Cayley table on which a binary operation is defined to act on the 123-avoidind and 132-avoiding pattern of AUNU permutation using a pairing scheme. The result have generate larger matrix from permutation of point of the AUNU Pattern of prime cardinality see 2. Some Basic Definition 2.1. B-algebra-: A B-algebra is a non-empty set  with constant 0 and binary operation “*” satisfying the following axioms. B1 ∗ =0 B2 ∗ 0= 