This master’s thesis investigates Birkhoff’s theorem, which classifies finite distributive lattices as lattices of order ideals of their join-irreducible elements. The main goal of the thesis is to explore possible generalizations of this fundamental result to broader classes of lattices, such as locally finite distributive lattices, general distributive lattices and finite semidistributive lattices. In this context, we introduce the concepts of order ideals and join-irreducible elements, showing how the key properties from finite lattices naturally extend to locally finite lattices with a least element. In the case of infinite distributive lattices join-irreducible elements are replaced by prime ideals and it is shown that a lattice is distributive if and only if it is isomorphic to a ring of sets. For the case of finite semidistributive lattices, we define reflexive relations that determine maximal orthogonal pairs of subsets of a given set. Using the so-called "kappa"-lattices, we demonstrate that a finite lattice is semidistributive if and only if it is isomorphic to the lattice of maximal orthogonal given by a factorization system without 2-cycles.
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