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ADVANCED ALGEBRAIC STRUCTURES M SURESH BABU Asst. prof. of Mathematics, Santhiram Engineering College, Nandyal, Kurnool District, Andhra Pradesh ABSTRACT: As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many -- perhaps most -- of these problems were in some way related to the theory of algebraic equations. Major themes include: Solving of systems of linear equations, which led to linear algebra Attempts to find formulae for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry Arithmetical investigations of quadratic and higher degree forms and Diophantine equations, that directly produced the notions of a ring and ideal. Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. This creates a false impression that in algebra axioms had come first and then served as a motivation and as a basis of further study. The true order of historical development was almost exactly the opposite. For example, the hyper complex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Most theories that are now recognized as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the history of group theory. INTRODUCTION: In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebra over a field. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their associated homeomorphisms, form mathematical categories. Category theory is a powerful formalism for analyzing and comparing different algebraic structures. Universal algebra is a related subject that studies the nature and theories of various types of algebraic structures as a whole. For example, universal algebra studies the overall theory of groups, as distinguished from studying particular groups. APPLICATIONS: Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The recently (As of 2006) proved Poincare conjecture asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem. In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian. IMPORTANCE: By abstracting away various amounts of detail, mathematicians have created theories of various algebraic structures that apply to many objects. For instance, almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associatively (to form semi groups); identity, and inverses (to form groups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory apply to rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. Mathematicians choose a balance between the amount of generality and the richness of the theory. Examples of algebraic structures with a single binary operation are: Magmas, Quasigroups, Monoids, Semi groups ,Groups More complicated examples include: Rings, Fields, Modules, Vector spaces, Algebras over fields, Associative algebras, Lie algebras, Lattices, Boolean algebras Keywords: Algebraic structures, Semi group, Group, Ring, Field, Lattice, k-module and Vectorial space. Notations to be remembered: 1. * a binary operator. 2. G a non empty set. 3. a, b, c ∈G elements of the non empty set G. 4. a * b closure law under the binary operator . 5. a *(b*c)=(a*b)*c Associative law under the binary operator . 6. a *e = e*a = a The existence of an identity under the binary operator Where, e is the identity element. 7. a *a-1= a-1*a = e Existence of inverses under the binary operator . where a-1 is the inverse element. 8. a *b = b*a Commutative law under the binary operator . Examples: 1) Show that the set of integers is a group under the operation of addition. Solution: i) Closure since sum of two integers is an integer, therefore I is closed with respect to addition. ii) Associatively : If a, b, c are any arbitrary elements in I, then (a+b)+c = a+(b+c) is true, because associative law of addition holds in I. iii) Existene of identity : The number 0 ∈ I, such that a+0 = a = 0+a for all Thus the integer 0 is the additive identity. iv) Existence of inverse : If a ∈ I then -a ∈ I such that a+(-a) = 0 (-a) +a. Thus every element in I has its additive inverse. Since the set I is satisfying all the properties of a group therefore, it is a group with respect to addition. Again, it is noted that addition of integers is commutative,i.e., a+b = b+a for all a, b ∈ I. Hence (I,+) is an abelian group of infinite order. 𝜔 2) Prove that the four roots of unity1, -1, i, i, where I = √−1 form an abelian multiplicative group. Solution: Preparing the compositon table for the set, G = {1, -1, i, -i} with multiplication composition as given below: X 1 -1 i -i 1 1 -1 i -i -1 -1 1 -i i i i -i -1 1 -i -i i 1 -1 The following properties are verified from the composition table: i)Closure Property: Since all the elements in the composition table are also elements of G, therefore, G is closed with respect to multiplication. Hence, multiplication is a binary operation on G. ii) Associatively Property: Ordinary multiplication is always associative. iii)Identity element: 1 is the identity element, because 1.1 = 1; 1.(-1) = -1; 1.(i)=I; 1.(-i) = i i.e., 1.a = a, for all a ∉ G. iv) Inverse Property: It can be verified from the composition table that 1.1 = 1; -1.(-1) +1; (-1).(-i) = I; i.(-1) = -i i.e., the inverse of 1,-1,i and –i are 1,-1, -i and i respectively. v) We see in the composition table that multiplication is commutative because all the entries in first, second, third and fourth rows in the table are identical to those of first, second, third and fourth columns respectively, i.e. the law a x b = b x a , for all a, b, ∈ G holds. Hence, G is an abelian group under multiplication. References: [1] Discrete and Combinatorial Mathematics, Fifth Edition, R.P. Grimaldi, B.V. Ramana, Pearson. [2] Discrete Mathematics Theory and Applications, D.S. Malik and M.K. Sen, Cengage Learning [3] J. L. Mott, A. Kandel, T.P. Baker, Discrete Mathematics for Computer Scientists and Mathematicians, Second edition 1986, Prentice Hall of India. [4] C.L. Liu, elements of Discrete Mathematics, Second Edition 1985, Mc Graw-Hill Book Company. Reprinted 2000. [5] Discrete Mathematics, Norman L. Biggs, Second Edition, OXFORD Indian Edition. [6] K.H. Rosen, Discrete Mathematics and applications, 5 th Edition 2003, Tata Mc Graw Hill publishing company. [7] Graph Theory with Applications to Engineering & Computer Science: Narsingh Deo, PHI ( 2004 ). [8] “ Discrete Mathematical Structures “ Jayant Ganguly, Sanguine. [9] Discrete Mathematics with Applications. Thomas Koshy, 2003 , Elsevier Academic Press. [10] Discrete Mathematical Structures with Applications to Computer Science, J.P. Tremblay and R. Manohar, 1975 , TMH.