2023/03/07 by Dale Worley, Worley, Dale R.
Computer Science · #Advanced Algebra and Logic #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2303.04267
Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice. This theorem can be extended as follows: A non-finite distributive lattice that is locally finite and has a 0 is isomorphic to the lattice of finite order ideals of the partial order of the join-irreducible elements of the lattice. In addition, certain ``well ordering'' properties are shown to be equivalent to the premises of the extended theorem.