Folding of Geometric Figures and Some Applications in Algebra

Abstract

This thesis sheds light on the concept of folding for some geometric figures, KU-graph of KU-algebra and KU-ideal of KU-algebra. The folding of a manifold is a one of the famous problems in the fields of geometry and topology. S.A.Robertson in [73] is the first one who introduced this idea, when he crumpled a sheet of paper in his hand and then crushed it flat against a desk-top and studied the stratification determined by the folds. Most folding and unfolding problems are attractive from a pure mathematical standpoint for the beauty of the problems themselves. However, most of the problems have close connections to important industrial applications. Paper folding has applications in sheet metal bending, packaging, and airbag folding. Unfolding polyhedra has applications in manufacturing, particularly sheet metal bending. Folding of graphs began with M. El-Ghouls work, in [22]. The conditional folding of manifold and a graph folding have been defined by El-Kholy [53, 54]. Unfolding of a manifold has been defined and discussed by M. El-Ghoul, in [20]. After that, many papers studied the folding, unfolding and deformation retraction of different types of manifolds; see [19, 21, 23, 24, 25, 28, 74, 77 and 78]. Abu-Saleem, in [76] introduced many results of some geometric transfers of the manifold on the fundamental groups. In this work, we firstly describe some geometric figures by using the algebra and we study the types of connectedness of geometric figures. Also, we give the effect of some geometric transfers (folding and unfolding) on the homology groups. During the past fifty years, the Mathematicians have defined different types of algebraic structures and they studied many mathematical applications, such as fuzzy sets, graph theory, n-fold and derivation. Summary . ii Imai and Iseki [35, 36] introduced the new two classes of abstract algebras, BCK-algebra and BCI-algebra. BCI-algebra is a generalization of BCKalgebra. Various Mathematicians have studied these algebras extensively and as a result a lot of literature has emerged. Iseki [37] introduced the concept of ideals in BCK-algebras. Chen [16] introduced n-fold positive implicative BCK-algebras. After that, Huang and Chen [33] introduced n-fold implicative BCK-algebras. Many Mathematicians have studied n-fold of some algebraic structures. For example, see [43, 58 and 84]. Prabpayak and Leerawat [69, 70] constructed a new algebraic structure which is called KU-algebra and they introduced the concept of homomorphism of KU-algebra. In this thesis, we introduce n-fold commutative, n-fold implicative and nfold positive implicative of KU-algebra as a natural generalization of KUcommutative, KU-implicative and KU-positive implicative of KU-algebra respectively. Also, we study n-fold KU-ideal, n-fold commutative ideal, nfold implicative ideal and n-fold positive implicative ideal of KU-algebra. Moreover, we obtained a few interesting properties, such as every n-fold KUpositive implicative and n-fold KU-commutative ideal is n-fold KUimplicative ideal. Similarly, the converse is also true.