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چكيده انگليسي :

Polar Graphs and Corresponding Involution Sets Loop and Steiner Triple System Zahra Bahrami z bahrami@math iut ac ir June 2011 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr S Ghahraman Taherian taherian@cc iut ac ir2000 MSC 51F05 51F20 Key words Involutorial di erence loop Involution set Polar graph A ne triple system Pseudo a ne space Abstract In this thesis we present an expanded account of Polar Graphs and Corresponding involutionsets Loop and Steiner Triple System by Helmut Karzel Silvia Pianta and Elena Zizioli 2006 Moreover we give some new concepts about permutation sets re ection structures and their relationswith geometric structures We describe shortly the concept of matriod The coordinatization ofthese matroids by a certain class of Moufang loops provides us with a powerful algebraic tool foranalyzing their matroid structure and relations in the coordinatizing loop give rise to con gurationtheorems in the matroid just as algebraic properties of elds give rise to the Desargues and Pappuscon gurations in projective eld planes We derive a partial binary operation from an involution set and we discuss if such operationis a Bol operation or a K operation We relate involution sets with loops We look for the pos sibility to construct loop nearrings by considering the automorphism groups of loops The mainconcern of this work is investigating the relation between special graphs and some algebraic struc tures A 1 factorization or parallelism of the complete graph with loops P E is called po lar if each 1 factor parallel class contains exactly one loop and for any three distinct verticesx1 x2 x3 if x1 and x2 x3 belong to a 1 factor then the same holds for any permutation of theset 1 2 3 To a polar graph P E there corresponds a polar involution set P I an idem potent totally symmetric quasigroup P a commutative with weak inverse property loop P of exponent 3 and a Steiner triple system P B It is shown that P E satis es the trapeziumaxiom I I I P is self distributive P is a Moufang loop P B is an a ne triple system and P E satis es the quadrangle axiom I 3 I P is agroup P B is an a ne space We describe a representation of any semiregular right loop by means of a semiregular bipartiteinvolution set or equivalently a 1 factorization i e a parallelism of a bipartite graph with at leastone transitive vertex In these correspondences Bol loops are associated on one hand to invariant regular bipartite involu tion sets and on the other hand to trapezium complete bipartite graphs with parallelism K loops orBruk loops are futher characterized by a sort of local Pascal con guration in the related graph We identify P with L Z2 via the bijection In the bipartite graph with parallelism we de ne operation and Therefore The followingstatements are equivalent 1 P E is a complete bipartite graph with parallelism 2 P E is a regular bipartite involution set 3 P is a loop 4 P is a loop 5 P is a loop We prove that a steiner triple system is equivalent with decomposition of complete graph to trian gle complete graph K3 also we prove a steiner triple system P B exists if only if with de ne aoperation P be a semisymetrice idempotent quasigroup 1

بهرامي، زهرا

گراف هاي قطبي و مجموعه هاي خودوارون نظير آنها، دورها و سيستم هاي سه تايي اشتاينر

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