I, EDEGHAGBA Eghosa Elijah, started my PhD studies 2013/2014 academic session at the University of Novi Sad, Republic of Serbia. Throughout the study with regards to program requirements, I am expected to undergo coursework and then to pass twelve (12) oral and written exams, totalling 120 ECTS credits and thesis of 60 ECTS credits. As at December, 2015 I have passed a total of eleven (11) of these examinations with GPA 8.91 and 110 ECTS credits. In January 2016, I am expected to pass the last exam. During the year, 2015, I gave two seminar talks; one in Universal Algebra, titled “On Mal'cev Conditions“ and the other in Lattice Theory, titled “Priestley Representation Theorem on Distributive Lattices“. So far, the feedback I got from the department of my progress has been an excellent one. Click here to read more.
Through the years, I have made a lot of progress toward the completion of my Ph.D. Alongside I have made significant headway into research that will be used toward my Thesis and I am on track to submit the application of the thesis proposal by the beginning of 2016 under the supervision of Prof. Dr. Branimir Seselja and is expected to be approved as soon as possible. It has been a very orienting study in which I have been engaged and introdued to serveral areas which will be very helpful for my thesis. I appreciated this experience very much, as it allowed me to have a solid understanding of the related work and theory of the field. This will help serve as a basis for my future research and help me to feel confident in my ability as a researcher and as an academician. Therefore, since I have pass almost all my exams but one I was then given the go ahead for research work and for publications. As at November, 2015, I along with my supervisor and co-supervisor sent a paper to Elsevier Journal for publication. The title of the paper is “Omega Lattices”. And presently I am already working on the second on which hopefully will be submitted for publication before the end of January, 2016. After the submission of this second paper, I will be starting off with the third, which already I have laid down the framework.
I am looking forward to this year, 2016, where I will continue along this trajectory and work toward the completion of my thesis in good time to graduate.
My thesis will deal with Omega algebraic and relational structures. The first research in this field started in eighties of the 20th century by Scott and Fourman. They introduced omega sets (Ω-sets), omega (Ω) represents a Heyting lattice. Their intention was to model intuitionistic and some other nonclassical logics.
By definition, an Ω-set is a pair (M, E), where M is a nonempty set and E is an Ω-valued equality M, which is defined as a symmetric and transitive Ω-valued equality on M. The latter (E) is a mapping from the square of the given set, M into a Heyting lattice, fulfilling a generalized transitivity. This mapping is acting as a multi valued equality. In the subsequent years this research was continued in other many valued logics, and further in the basic research of Fuzzy set theory (by Belohlavek, Demirci, Hohle, Seselja, Tepavcevic and others). In the present research, omega sets will be equipped with some basic algebraic and relational structures, and the investigations will be focussed on universal algebraic aspects of the new omega algebras and relational structures. They are supposed to fulfil various identities, transformed into lattice-theoretic formulas. In the other hand, omega relations will also be introduced, by which omega ordered structures (like omega lattices) will be investigated. General properties on all these are that particular quotient structures over so called cuts, are classical algebras, like groups, lattices etc.
This topic is new, connecting universal algebra with lattice valued (fuzzy) structures, and I expect to obtain valuable results in this field. First results have already been obtained in the paper about omega lattices, recently submitted.
EDEGHAGBA Eghosa Elijah