14665

13207

كارشناسي ارشد

رياضي محض

اصفهان : دانشگاه صنعتي اصفهان

1398

پنج، 91ص.:نمودار

واژه نامه

1398/03/13

كتابنامه

رياضي محض

رياضي

1398/03/18

2539381

Expansions of the Ordered Additve Group of Real Numbers by two Discrete Subgroups Masumeh Rezaei masumeh rezaei@math iut ac ir May 29 2019 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mohsen Khani mohsen khani@cc iut ac irAdvisor Dr Mojtaba Aghaei aghaei@cc iut ac ir2000 MSC 03F52 03B42Keywords Model theory continuous fractions Ostrowski expansion interpretation decidablilityAbstract This M Sc thesis is based on the following paper Expansions of the ordered additive group of real numbers by two discrete sub H Pgroups The Journal of Symbolic Logic 81 3 2016 1007 1027 One of the important concepts in logic and model theory is decidablility The purpose of this thesisis to prove the decidability of the structure Ra R Z Za where a is quadratic Z is apredicate for the integers Za a predicate for the set an n Z In order to be able to prove that Ra is decidable we interpret it within a decidablestructure B N P N sN which is known by a result of B chi to be decidable The structure B is called two sorted due to two sets N and P N In the structure B the function sNis the successor function in the form sN N N such that every member x in N successor is sentto x 1 and the relation is a relation on N P N such that t X iff t X Interpreting means that we define the corresponding with sets relationships and operations within thestructure Ra definable sets inside the structures B To interpret the structure Ra within the structure B we first create a set A P N a set of sets Nand a set Af in A which are to play the same role the same R and N We then define the sum and order of these two sets then we introduce the mapZ Af in Z N by which we define the set N within the structure B In the next step we introduce the map O in the form O A O I 1 by whichwe define the set R within the structure B In this way we impute a set to any number and define sumand order on the sets Then we introduce the map S in the formS A 1 1 0 1 mod 1 by which we change the interval I to 0 1 so that wecan interpret all the real numbers within the structure B Notice that with the map Z we interpreted the set N and with the map S we interpreted the set Rwithin the structure B To be able to synchronous interpret these two sets within the structure B wefirst define the set B Af in 0 1 2 including regular pairs The set 0 1 2 means that thereare several natural numbers inside the interval aZ X a Z X 1 Then by definition sum andorder on B we introduce map R B B B N In the final step to synchronous interpret two sets N and R we put the set C A B then define sumand order on C as well we define A C and B C which play respectively the role of Na andN And at the end we introduce the mapT C C B B A R 0 N Na We get the map T that T B N and T A Na So by map T interpret structure

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