A  short review of decidability of boolean algebras ‎and‎  structure‎‏e of rational numbers ‎in different languages

Zahra Sheikhaleslami; Saeed Salehi

doi:10.53730/ijhs.v6nS7.11572

Authors

Zahra Sheikhaleslami Ph.D. Student of Tabriz Uni¬versity. The University of Tabriz P.O.Box 51666- 16471,Bahman 29thBoulevard; Tabriz; Iran.
Saeed Salehi Research Institute for Fundamental Sciences (RIFS),University of Tabriz, 29 Bahman Boulevard, P.O.Box 51666–17766, Tabriz, Iran & School of Mathematics, Institute for Research in Fundamental Sciences (IPM),P.O.Box 19395–5746, Tehran, Iran.http://saeedsalehi.ir/

Keywords:

Boolean algebras, Decidability, Model Theory, Quantifier-Elimination

Abstract

This article consists of two parts.First,‎‎we study boolean algebras.Boolean algebras are famous mathematical structures.Tarski showed the decidability of the elementary theory of Booleana lgebras.In this paper, we study the different kinds of Boolean algebras and their properties. And we present for the first-order theory of atomic Boolean algebras a quantifier elimination algorithm. The subset relation is a partial order and indeed a lattice order,And I will prove that the theory of atomic Boolean lattice orders is decidable, and furthermore admits elimination of quantifiers.‎So the theory of the subset relation isdecidable‎.‎‎And ‎we ‎will ‎study decidability of atomlss boolean algebra.Second part,‎of this paper we sho‏w that the structure of rational numbers in different languages has the property of ‎quantifier ‎elimination,and hence is decidable.This proofes are organized in two‎ parts.

Downloads

Download data is not yet available.

References

Boolos, George S. Burgess, John P. Jeffrey, Richard C, Computability and Logic,Cambridge University Press (5th ed. 2007), ISBN: 9780521701464.

Burris, Stanley N., and H.P. Sankappanavar, CA Course in Universal Algebra,1981. Springer-Verlag. ISBN 3-540-90578-2 Free online edition.

Cegielski, Patrick; The Elementary Theory of the Natural Lat¬tice is Finitely Axiomatizable, Notre Dame Journal of Formal Logic 30:11:988138-150.

Christopher C Leary, Lars Kristiansen; A Friendly Introduction to Mathematical Logic, Milne Library, 2019.

Clara Trullenque Ortiz, COMPLETE THEORIES OF BOOLEAN ALGEBRAS,Departament de Matematiques Informatica,June 27, 2018.

D. Prawitz, Dag Westerstahl, Logic and Philosophy of Science in Uppsala, Springer Science Business Media, Tir 8, 1392 AP - Philosophy - 614 pages. .

Dutilh Novaes, Catarina; Formal Languages in Logic,November 2012, ISBN: 9781107020917.

Enderton, Herbert B, A Mathematical Introduction to Logic, Academic Press (2nd ed. 2001) , ISBN: 9780122384523.

Givant Steven and Halmos Paul , Introduction to Boolean algebras, Undergraduate Texts in Mathematics. Springer, 2009

https://en.wikipedia.org/wiki/Mereology

https://plato.stanford.edu/entries/mereology/notes.html1

https://www.oxfordbibliographies.com/view/document/obo- 9780195396577/obo-9780195396577-0313.xml

Monk, J. Donald; Mathematical Logic, Springer (1976), ISBN: 9780387901701.

Salehi, Saeed, On Axiomatizability of the Multiplicative Theory of Numbers, Fundamenta Informaticae ,159:3 (2018) 279-296. DOI: 10.3233/FI-2018-1665

Salehi,Saeed,Axiomatizing Mathematical Theories :Multiplica- tion, in:A.Kamali-Nejad (ed). Proceedings of Frontiers in Math¬ematical Sciences ,Sharif University of Technology .Tehran , Iran 2012 , pp.165-176.

Smorynski, Craig; Logical Number Theory I, Springer-Verlag (1991).

Steven Givant, Paul Halmos, Introduction to Boolean Alge- bras,Springer (2009)

Suryasa, I. W., Rodríguez-Gámez, M., & Koldoris, T. (2021). Health and treatment of diabetes mellitus. International Journal of Health Sciences, 5(1), i-v. https://doi.org/10.53730/ijhs.v5n1.2864