Henry Rose

Varieties of Lattices

In this chapter we discuss some of the more recent results and give a general overview of what is currently known about lattice varieties. Of course it is impossible to give a comprehensive account. Often we only cite recent or survey papers, which themselves have many more references. We would like to apologize in advance for any errors, omissions, or miscrediting of results.

Elementary equivalent pairs of algebras associated with sets

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.

Absolute Retracts as Reduced Products

We show that in finitely generated congruence distributive varieties every absolute retract is a product of reduced powers of maximal subdirectly irreducibles.

Partition Complete Boolean Algebras and Almost Compact Cardinals

For an infinite cardinal K a stronger version of K-distributivity for Boolean algebras, called k-partition completeness, is defined and investigated (e. g. every K-Suslin algebra is a K-partition complete Boolean algebra). It is shown that every k-partition complete Boolean algebra is K-weakly representable, and for strongly inaccessible K these concepts coincide. For regular K ≥ u, it is proved that an atomless K-partition complete Boolean algebra is an updirected union of basic K-tree algebras. Using K-partition completeness, the concept of γ-almost compactness is introduced for γ ≥ K. For strongly inaccessible K we show that K is K-almost compact iff K is weakly compact, and if K is 2K-almost compact, then K is measurable. Further K is strongly compact iff it is γ-almost compact for all γ ≥ K.

Amalgamation in the pentagon variety

The amalgamation class Amal (N) of a lattice variety generated by a pentagon is considered. It is shown that Amal (N) is closed under reduced products and therefore is an elementary class determined by Horn sentences. The above result is based on a new characterization of Amal (N). The lattice varieties whose amalgamation classes contain Amal (N) as a subclass are considered.

ULTRA-UNIVERSAL MODELS

Abstract The concept of ultra-universal algebras in varieties is generalized to models of first order theories. Characterizations of theories which have ulta-universal models are found and general examples of ultra-universal models are investigated. In particular we show that a theory has an ultra-universal model iff it is consistent and its class of models satisfies the joint embedding property.

Rudin-Keisler Posets of Complete Boolean Algebras

The Rudin-Keisler ordering of ultrafilters is extended to complete Boolean algebras and characterised in terms of elementary embeddings of Boolean ultrapowers. The result is applied to show that the Rudin-Keisler poset of some atomless complete Boolean algebras is nontrivial.

Varieties with cofinal sets: examples and amalgamation

A variety V has a confinal set S⊂V if any A∈V is embeddable in a reduced product of members of S. Amalgamation in and examples of such varieties are considered. Among other results, the following are proved : (i) every lattice is embeddable in an ultraproduct of finite partition lattices ; (ii) if V is a residually small, congruence distributive variety whose members all have one-element subalgebras, then the amalgamation class of V is closed under finite products