The thesis develops functorial semantics for algebraic and regular logic. The first part starts by briefly presenting category theory, then the concept of an algebraic theory is introduced as a special case of a first order logic theory, in which you only have equations and operations. The classical notion of a model is expanded to categories in which such a theory can be expressed. For each algebraic theory we may define a special syntactic category, which represents it. It turns out that you can uniquely identify each model of such a theory with a functor that preserves the structure of the syntactic category. This is expressed in the form of an equivalence of categories. With the help of this equivalence a duality between syntax and semantics is explored. The second part begins with the description of a class of categories called regular categories and the motivation for their definition in terms of examples and nice properties that these categories posses. An extension of the simple algebraic logic is then developed into the so called regular logic which besides equations and operations includes relation symbols, the truth constant, conjunction and the existential quantifier. This makes the logic more rich and makes it possible to express concepts like the image of a morphism. Analogous with the first part we define the syntactic category of a regular theory, with the help of which you can show an equivalence between models of a regular theory and functors that
preserve regular structure.
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