Lattice theoretic notions

Abstract

In this chapter the basic notions of lattice theory are collected. Besides the definition of a lattice the important definitions are the distributivity, the modularity and the orthomodularity of a lattice (Definitions 3.5, 3.6 and 3.10). The proposition stating that a lattice on which a finite dimension function exists is necessarily modular (Proposition 3.3) will become important in the Chapter 6. The one-to-one correspondence between prime filters in a lattice and lattice homomorphisms on the lattice into a Boolean lattice (Proposition 3.11) will be used later to show that there exist no lattice homomorphisms from a quantum logic into a Boolean lattice. The notion of a partial algebra, partial Boolean algebra and partial algebra homomorphism will come up in Chapter 9 naturally in connection with a certain concept of hidden variable theory of quantum mechanics.