Ockham Algebras And Their Congruences

In this thesis we ?rstly study GBωalgebras. A GBωalgebra (L; f) is anOckham algebra, where ?x∈L, there exists n≥0 such that fn(x) and fn+1(x)are complementary. We characterized the structure of the lattice of congruenceson such an algebra (L; f). In particular we show that the lattice of congruenceson L is Boolean if and only if L is a ?nite Boolean algebra. We also give asu?cient and necessary condition for the subdirectly irreducible GBωchains.Then we study the principal congruences on S1-algebras. The class of S1-algebras is a subclass of GBωalgebras, and it includes Boolean algebras andStone algebras. We show that an S1 algebra has the principal congruenceproperty if and only if it is ?nite, the determination congruence is principal,and every determination class has the same property. We also ?nd all GBnchains which have the principal congruence property.And then we look for those bpO algebras on which all congruences are per-mutable. bpO algebras are the mixture of Ockham algebras and pseudocom-plemented algebras. We give a characterization of the permutable congruenceson bpO algebras via Priestley duality.Finally, we study the lattices whose Hass diagrams are regular graphs, namedregular graph lattices. We show that a ?nite distributive lattice is a regulargraph lattice if and only if it is a Boolean lattice. We also ?nd out all the1-degree graph lattices and 2-degree graph lattices. In particular, we show thatthe 8-element Boolean lattice is the smallest 3-degree regular graph lattice.