On Pseudo-BCI algebras

In this thesis, definition and basic properties of BCI-algebras, atoms, branches, medial and ideals of pseudo-BCI algebras and p-semisimple pseudo-BCI algebras are investigated. In the first section, the subject of the thesis is introduced, the goal of working on this area and the development process of the subject are summarized. In the second section, by primiraly considering the paper of Georgescu and Iorgulescu which is published in 2001 , the paper of Dymek which is published in 2012, the paper of Jun, Kim and Neggers which is published in 2006, the paper of Hao and Li which is published in 2004 and the book of Hungerford which is published in 1974, some definitions and remarks are given which are necessary for easy understanding of next sections. Also, the connection between BCK-algebras and BCI-algebras is given. In the third section, by primiraly considering the paper of Jun, Kim and Neggers which is published in 2006, the paper of Dudek and Jun which is published in 2008, the paper of Lee and Park which is published in 2009 and the paper of Dymek which is published in 2012, definition and some basic properties of pseudo-BCI algebras are given and the difference between pseudo-BCK algebras and pseudo-BCI algebras is investigated. In the forth section, by primarily considering the paper of Dymek which is published in 2012 and the paper of Jun, Kim and Neggers which is published in 2006, four subtitles which are atoms, branches and medials of pseudo-BCI algebras and p-semisimple pseudo-BCI algebras are investigated. In this section, the relations of these structures with each other are mentioned. Also, the question which is "When does a pseudo-BCI algebras turn into a group?" is answered. In the fifth section, by primiraly considering the paper of Lee and Park which is published in 2009, the paper of Dymek which is published in 2012 and the paper of Jun, Kim and Neggers which is published in 2006, definition and some basic properties of pseudo-BCI ideals are introduced. It is investigated when a pseudo-BCI subalgebra is a pseudo-BCI ideal. In the sixth section, the conclusion of this work and the future work are given. Key Words: BCK-Algebras, BCI-Algebras, Pseudo-BCK Algebras, Pseudo-BCI Algebras.