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datum odbrane: 25.10.2011.

Osnove proučavanja Bulovih funkcija i jednačina su postavili Boole, Schröder i Löwenheim u drugoj polovini devetnaestog i početkom dvadesetog veka. Ova oblast se intenzivno razvija u drugoj polovini dvadesetog veka. Taj razvoj se uglavnom odvija u dva osnovna pravca: u pravcu specijalizacije i pravcu generalizacije. Kada se govori o specijalizaciji, misli se na proučavanje pojedinih vrsta Bulovih funkcija i jednačina, kao što su proste Bulove funkcije i jednačine, vrednosne (“swit- ching”) funkcije i jednačine, Bulove diferencijalne jednačine, zatim na proučavanje Bulovih jednačina napojedinim vrstama Bulovihalgebri (relacionealgebre, kompletne Bulove algebre), kao i wihovu primenu u raznim oblastima: u detekciji grešaka u logičkim mrežama, teoriji kodiranja, teoriji automata, teoriji grafova, optimizaciji itd. Kada se govori o generalizaciji misli se na uopštavanje poznatih rezultata o Bulovim jednačinama na jednačine na drugim mrežama, kao što su ograničene distributivne mreže, pseudokomplementarne distributivne mreže, Stonove algebre i dr. Prirodno uopštenje Bulovih jednačina su jednačine na viševrednosnoj logici. Aksiomatizacijom algebre koja odgovara Postovoj viševrednosnoj logici (nazvane Postovomm algebrom) otvara se novo poqe generalizacije -- Postove jednačine. I rezultati Izloženi u ovom radu predstavljaju generalizaciju nekih tvrđenja o Bulovim jednačinama na jednačine na Stonovim algebrama, viševrednosnoj logici i Postovim algebrama

This doctoral dissertation belongs to the scienti¯c discipline Algebra and logic. General solutions of an equation are present in various ¯elds of mathematics. Especially, the general solutions were extensively studied in Boolean algebras. In this doctoral dissertation some known results about Boolean equations are generalized to equations on Stone algebras, equations on multiple-valued logic and to equations on Post algebras. The dissertation consists of ¯ve chapters divided in sections, Appendix and Ref- erences. In Introduction some basic notations which will be used in next chapters are given. Because of many theorems from this ¯eld represent generalizations of the cor- responding results for Boolean equations, main results on Boolean functions and equations are exposed in Chapter 2. In Chapter 3, assuming that a general solution is known, the class of reproductive general solutions of the equation f(x1; : : : ; xn) = 0, where L is a Stone algebra and f : Ln ! L is the function with substitution property, is described. All general solutions of equations in one variable on multiple-valued logic are described in Chapter 4. S. Rudeanu in [41] determined the most general form of the subsumptive general solution of a Boolean equation. He also proved that every Boolean transformation was the parametric general solution of a consistent Boolean equation. He stated an open problem: extend this research to Post algebras. Chapter 5 contains some results related this problem. A necessary and su±cient conditions for the existence of a Post function f : Pn ! P (P is a Post algebra) such that the given set of reccurent inequalities be the solution of equation f(x) = 0, are given. We also proved that every Post transformation was the parametric solution of some consistent Post equation.

Algebra, logika

512

monograph text - theses Text