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International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, January-2016 Modelling Group Structures Faisal Hussain Nesayef Department of Mathematics, Faculty of Science, University of Kirkuk, Iraq ABSTRACT Axiomatic structures have been studied by various mathematicians, such as [1] and [2]. The formation of axiomatization of a commonly known kind of algebraic structure such as groups have steadily established. The consequences of basic terminologies are identified in the field of group theory. Keywords Axiomatization, Definable sets, Embedding, Homomorphism, Interpretation, Reducts, Sentence, Signature, Structure, Ultraproducts. INTRODUCTION In this paper we consider group in the signature of addition and multiplication structures, ie ( 0, 1; +, . ) and ( 0, 1; -, . ), where 0 and 1 are the constant symbols,"β " is the unary function symbol and " + "and " . " are binary functions. To achieve our objectives, we started with some basic principles enabling us to progress through the whole of our process in this research. 1. BASICS These basic definitions and properties are given in [1], [2], [3] and [4]. Definition 1.1: A signature πis a system (3-tuples ) , ( C,F,R, π ), which consists of a set C of constant symbols, a set F of function symbols a set R of relation symbols and a signature symbol π βΆ πΉ βͺ π β β\{0}. Example 1.1 π β‘ Function symbol +, . , β β‘ Arity 2 β₯ β‘ Relation symbol 0,1 β‘ Constant symbol < π ; + , . ,β, β₯ ; 0 ,1 > Definition 1.2 : A sign structure β³ is a quadruple ( M , C , F, R ) which consists of, (β βΆ π β β ), πΉ = ( π β πΉ) and π βΆ π β β), where π β πΉ and R is a π(π ) β ππππ‘π¦ relation is M. Example 1.2: Consider π = ( 0, 1 ; + ; . ; <), where π (+) = π(. ) = π(<) = 2. Then (π ; 0,1 ; +, . ; < ) is an ordered ring structure. Take π = < . , 1 , (β1)β1 > is a π β structure for a group. Definition 1.3: A homomorphism of π β structure fromπto β¬ is a Relationπ: π β β¬ such that 1. for all relation symbols R of π , if ( π1 , β¦ , ππ ) β π π π‘βππ π(π1 ), β¦ , π(ππ ) β π β¬ . 2. For all function relation symbols of π. π (π π (π1 , β¦ , ππ ) = π β¬ ( π(π π (π1 ) , β¦ , π(ππ )) 3. For each constant symbol C of π , (π π ) = π β¬ Page | 1 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, July-2016 An embedding is a homomorphism which is injective and for all relation symbols R, (π1 , β¦ , ππ ) β π π ππ πππ ππππ¦ ππ π(π1 ), β¦ , π(ππ ) β π β¬ . An isomorphism of π β structure is a homomorphism π βΆ π β β¬ if there is π: π β β¬ with π. π the identity on β¬. ie, π. π is the identity. Lemma 1.1: Every isomorphism of π β structure is an embedding. Proof : Suppose π βΆ π β β¬ is an isomorphism and R is a relation symbol of π. (π Suppose πΜ = 1 , β¦ , ππ ) β π΄ , and (π) β π β¬ . Then writing π β1 for inverse of π , we have π β1 (π(π)) β π π . So πΜ β π π . Therefore π is an embedding. Example 1.3: There is an inclusion map < π , + , . , β , 0 , 1 , β€ > β < β, +, . , β, 0 , β€ >. 2. PRODUCTS AND EXPANSIONS The structures < π; + > and < π; + , . > are different as they have different signatures. We say that < π; + > is a reduct of < π; + , . >. Also < π, + , . > is an expansion. Products and Ultra products Definition 2.1: Let I be a non-empty set and ( β³π ) iβI be a family of a structures. The productβ³ = ππβπΌ β³π is π β structure defined by: 1. The domain of β³ is a set of functions πΌ = πΌ β ππβπΌ β³π π π‘ πΌ(π) β β³π , π β πΌ (πΌ(π))πβπΌ = πΌ 2. For each constant symbol π ππ ππΆ β³ = (πΆ β³π )πβπΌ 3. For π, π π symbol , anity n and πΜ β β³ π , π β³ (πΜ ) = π β³π (πΜ (π)), π€βπππ π β I 4. For π relation symbol anity n and πΜ = (π1 , β¦ , ππ ) β β³ π πΜ β π β³ β πΜ (π) β π β³π β π β³π , β π β πΌ . Example 2.1: Consider a family ( πΊπ ) π β πΌ of abelian groups. π = < + , β product of abelian groups. For π β π’ the support is defined by: Supp(g) ={π β πΌ = π’(π) β 0}. Lemma 2.1: , 0 >. Let π’ = π±π βπΌ πΊπ , the π» = { π β π’ = π π’ππ(π) ππ ππππ‘π }.H is a subgroup ofπ’ . Proof : Supp (π π’ ) = π π’ππ (π π’ )π β πΌ = β . π π’ β π» β π β π». ππ’ππ (βπ) = π π’ππ(π), π π’ππ(π + π) β π π’ππ(π) βͺ π π’ππ(π). If a, π β π» , then βπ πππ π + π β π» πβπππππππ π» is a subgroup of π’ . π» is called the direct sum of the π’π , written π» =βπβπΌ π’π As π’ is abelian, π» is a normal subgroup of π’.So the quotient π’/π» is a group. An element of π’/π» is an equivalence class of elements of π’ under the equivalence relation π~π if and only ifβπ β π» . That means π, π are the same class except possibly at finitely many indices. Convention: If it does not confuse we use the same letter for a structure as its domain e.g π’, πΊ and β³ πππ β³. In π’/π» π1 + π» = π2 + π» if and only ifπ1 β π2 is finite, if and only ifπ1 (π) = π2 (π) except for finitely many indices. 3. FORMAL LANGUAGES AND THEIR INTERPRETATIUON We want to see new to build a formal first order language πΏ(π) and the signature π from a π_ structure. We can use πΏ(π) to describe π_ structures . Terms: We have an infinite supply of variables. We use letters π₯, π¦, π§, π₯1 , π₯2 , β¦ ππ‘π to represent variables. The set of terms of πΏ(π) is defined recursively i. Every variable is a term. ii. Every constant symbol of π is a term. iii. If f is a function symbol of π, of arity n , and π‘1 , β¦ , π‘π is a term, then f ( t1 β¦tn ) is a Term. iv. Only things build from i,β¦,iii is finitely many steps are terms . Page | 2 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, July-2016 Example 3.1: π= <π Function system .1 ,π , π n symb.2 π> Constant symbol π, π₯, π (π), π(π₯, π), π(π(π , π) , π (π)π₯ are terms. Note: often we use a binary function symbols like +, . and write π‘1 + π‘2 instead of +(π‘1 , π‘2 )Similarly we write π₯ β1 instead of π(π₯) . Terms are just strings of symbols without meaning. Note that terms of πΏ(π) are called π_π‘ππππ or πΏ_π‘ππππ or,(π)_π‘ππππ . Interpretation of Terms π_π‘ππππ can be interpreted as functions in π_π π‘ππ’ππ‘π’πππ . Example 3.2: π =< π ; + , . , β , 0 , 1 > The term [ ((π₯. π₯) + π₯) + 1 ] can be interpreted as the polynomial functions β β β π₯ β π₯2 + π₯ + 1 Let π be a π_ππ‘ππ’ππ‘π’ππ , toπ_ π‘πππ π‘(π₯Μ ) suing variable from π₯Μ = (π₯1 , β¦ , π₯π ), we associate a function π‘ π : π΄π β π΄ as follows i. If π‘ = π₯π then π‘ π (π) = ππ . ii. If π‘ = π then π‘ π = π π . iii. If π‘ = π(π‘1 , β¦ , π‘π ) then π‘ π (π₯Μ ) is given by composition: π‘ π (π₯Μ ) = π π (π‘ π (π₯Μ ), β¦ , π‘ππ (π₯Μ )) Formulas We now define πΏ(π) β πππππ’πππ ( πΏ β πππππ’πππ ), which are interpreted as statements about the structure: i. If π‘1 and π‘2 are π_ π‘ππππ , then π‘1 = π‘2 is an πΏ(π) formula . ii. If π‘1 , β¦ , π‘π are π_ π‘ππππ and R is a relation symbol of arity n, then π (π‘1 , β¦ , π‘π ) is an πΏ(π) β πππππ’ππ. iii. If β , π are πΏ(π) β πππππ’πππ , then (β β§ π) and β β are πΏ(π)_πππππ’ππ. iv. If β is an πΏ(π) β πππππ’ππ and x is a variable, then β π₯ β as an πΏ(π) β πππππ’ππ . v. Only something built from (i) β (iv) in finitely many steps is an (π) β πππππ’ππ. Formulas from (i) and (ii) are called atomic formulas. Those built from (i) , (ii) and (iii) are quantifiers β free formulas. We define the language πΏ(π) to be the set of all(π) β πππππ’πππ . πΏ(π) is called a first β order language because we only quantify over elements, not over a set of elements. Theorem 3.1 Letπππππ = < +, . ; β, 0, 1 > be the structure of the ring of integers considered as a πππππ structure. Then every polynomial function in Z [x1, β¦., xn ] is the interpretation I Z of a πππππ term. Proof 0,1, -1 are polynomials. So is xi . If p and q are polynomials then p + q is a polynomial. pq is also a polynomial. Now 0, 1 and -1 are the interpretations of the terms written in the same way. Similarly variables xi are terms. Suppose p and q are polynomial functions and assume that there are terms Ο and Ο such that Ο = p and π = q Then the terms (Ο+ π ) and Οπ have interpretation in Z which are p +q and p q respectively. By induction every polynomial is the interpretation in Z of a πππππ term. Theorem 3.2: Let M β¨Ts where Ts is the theory of the successor. Suppose a, b βM \ N. Then there is an automorphism Ο of M such that Ο ( a ) = b. Proof Assume that M = M1 for some I, as every symbol is isomorphic to Mi for some i. Since a, b β M \ N, we have a = ( i, n ) and b = ( j, m ) for some i, j β I and some n, m β Z. x βΌy if and only if x = s ( y ) for some r β Z. Page | 3 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, July-2016 If x βΌy define Ο by Ο ( x ) = x for every x such that: a βΌ b and Ο ( i, r ) = ( i, r + m βn ), and Ο ( j, r ) = ( i, r ) Both cases show that Ο ( a ) = b and that Ο is an automorphism. ACKNOWLEDGMENT The author would like to acknowledge the University of Kirkuk for providing research facilities in the department of mathematics. The author also thanks the reviewers for reading this paper. Thanks are also due to the International Journal of ERP for publishing this work. REFERENCES [1]. [2]. [3]. [4]. P. Rothmale, "Introduction to Model Theory", Algebra, Logic, Applications Series, Volume 15, 2000. J. J Rotman, "The Theory of Groups, An Introduction", second Edition, Allyn and Bacon, Inc, 1973. M.Drosle and R. Gobel, "Advances in Algebra and Model Theory", C R C 1998. W. Felscher, "Lectures in Mathematical Logic", Algebra, Logic, Applications Series, Volume 1, 2000. Page | 4