An Ockham algebra is a natural generalization of a well known and important notion of a boolean algebra. Regarding the latter as a bounded distributive lattice with complementation (a dual automorphism of period 2) by a dual endomorphism that satisfies the de Morgan laws, this seeminglymodest generalization turns out to be extemely wide. The variety of Ockham algebras has infinitely many subvarieties including those of de Morgan algebras, Stone algebras, and Kleene algebras. Folowing pioneering work by Berman in 1977, many (...) papers have appeared in this area oflattice theory towhich several important results in the theory of universal algebra are highly applicable. This is the first unified account of some of this research. Particular emphasis is placed on Priestly's topological duality, which invloves working with ordered sets and order-reversing maps, hereby involvingmany problems of a combinatorial nature. Written with the graduate student in mind, this book provides an ideal overview of this are of increasing interest. (shrink)

We identify the \-ideals of a distributive demi-pseudocomplemented algebra L as the kernels of the boolean congruences on L, and show that they form a complete Heyting algebra which is isomorphic to the interval \ of the congruence lattice of L where G is the Glivenko congruence. We also show that the notions of maximal \-ideal, prime \-ideal, and falsity ideal coincide.

In this note we shall show that if L is a balanced pseudocomplemented Ockham algebra then the set ${\fancyscript{I}_{k}(L)}$ of kernel ideals of L is a Heyting lattice that is isomorphic to the lattice of congruences on B(L) where ${B(L) = \{x^* | x \in L\}}$ . In particular, we show that ${\fancyscript{I}_{k}(L)}$ is boolean if and only if B(L) is finite, if and only if every kernel ideal of L is principal.