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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.

Tema ove disertacije je involucijau algebarskim strukturama. Involucije su bijektivna preslikavanja koja se poklapaju sa svojim inverznim funkcijama. One se pojavljuju u skoro svim oblastima matematike: podsetimo se samo projektivne geometrije, teorije algebarskih krivih, inverzije u euklidskoj geometriji i njenog značaja za modele hiperboličke geometrije, teorije matrica i drugih disciplina. Cilj disertacije je da prikaže teoriju involutivnih algebri, tj. neke rezultate u okviru te teorije. Najviše su istraženi odnosi između algebarskih zakona i involucije, i ti odnosi daju jednu sasvim novu algebarsku teoriju. Materijal je podeljen u četiri dela. U prvom delu se posmatraju tzv. Plonkine sume. Ispostavilo se da su mnoge klasične konstrukcije u algebri samo specijalni slučajevi Plonikih suma. Kako bismo ih prilagodili izučavanju involutivnih algebri, ove sume su modifikovane, tako da dobijamo involutivne Plonkine sume. U radu su ispitane neke osobine takvih suma. U drugom delu istražujemo involutivne polugrupe. Između ostalog, dokazano je da je klasa regularnih *-traka globalno određena. Treći deo prikazuje neke od najnovijih rezultata u oblasti involutivnih poluprstena. Najzad, poslednji, četvrti deo govori o involutivnim prstenima. Posmatrani su neki poddirektno nesvodljivi prsteni sa involucijom, i dokazan je involutivni analogon čuvene teoreme N. Jacobsona., The topic o f this dissertation is involutionin algebraic structures. Involutions are bijective mappings which coincide with their inverse functions. They appear in almost all mathematical disciplines: recall projective geometries, theory of algebraic curves, inversion in euclidean geometry and its importance in the models of hyperbolic geometry, theory of matrices and other parts of mathematics. The aim of this dissertation is to present the theory o f involution algebras, i.e. some results in the frame o f that theory. We are investigating the relationship of algebraic laws and involution, which together give a new algebraic theory. The material is divided into four parts. In the first part, we are considering the so called Plonka sums. It turned out that many classical constructions in algebra are special cases o f Plonka sums. We modify these sums in order to make them applicable to involution algebras, and so we obtain the involutorial Plonka sums, whose properties are explored. In the second part, we investigate involution semigroups. Among other things, it is shown that the class o f regular ‘ -bands is globally determined. The third part is about semirings with involution. We review some of the latest results in the area o f involution semirings. The final, fourth part is about rings with involution. We are considering some subdirectly irreducible involution rings and prove an involutorial analogue of the wellknown theorem o f N. Jacobson.