关于Boole格同态的扩张

EKTENSIONS OF HOMOMORPHISM IN BOOLEAN LATTICE

本文主要证明了Boole格到格上的同态Φ可扩张为的完备化到的完备化上的备同态的充分必要条件是:Φ的核是一个分划。

This paper gives the following basic Theorem:If Φ is a homomrphism of the Booleau lattice onto the lattice, the kernel of which is a cut, then there exists one and only one complete homomorphism Φ from the completion of onto the completion of, such that Φ=Φ in.Definition. Suppose that Φ is a homomorphism of the Boolean lattice onto the lattice,then Φ is said completible if and only if a subset S′={Φ(x)|x∈S} of has the least upper bound or the greatest lower bound which equals when a subset S of has the least upper bound or the greatest lower bound.And this paper gives next the following Theoem:Let Φ be a homomorphism of the Boolean lattice onto the lattice then the kernel of Φ is a cut if and only if Φ is completible.