Homomorphisms and modular functionals

Abstract

This paper is concerned with complemented modular lattices containing the elements 0 and I. The first part treats of homomorphisms of the lattice L, their existence, determination and invariant properties. The second considers norms (i.e., sharply positive or, alternatively, strictly monotone modular functionals) and quasi-norms (i.e., positive or monotone modular functionals) on L, their interconnections, and necessary and sufficient conditions for their unicity up to linear transformations. There are six main parts to the paper, as follows: 1. The homomorphism theorem. The dual concepts of a--ideal and r-ideal are defined for general lattices. Duality is essential throughout the paper. The C-operator, which takes all elements of a subset of L into their complements, is introduced, and C-neutral ideals are defined as those which appear in complementary pairs a, a. Theorems 1 and 2 state that any C-neutral pair of ideals determine a congruence in L by means of any one of six equivalent conditions. These conditions are recognizable as those appearing in Boolean algebra, but the proof of their equivalence in the general case considered here is far from trivial, since it requires the fundamental Lemma 7. Theorem 3 states that all congruences are thus obtained from C-neutral ideals. Quotient lattices L/a are defined, and it is obvious that every homomorph of L is equivalent to an L/a. For example, consider a regular Caratheodory measure in a metric space, the measure of the space being 1. In the Boolean algebra of measurable sets, the sets of measure 0 and measure 1 are complementary C-neutral ideals, the first a-, the second 7r, and the quotient lattice is isomorphic with a sublattice of the Gs's. 2. The preservation of normal ideals under homomorphism. The operators c,, cX, and ', are defined. By means of the first two we define normal ideals, the upper and lower segments of MacNeille's cuts, whose main reason for existence is to make up for the gaps when L is not complete. The main theorem (Theorem 7) states that a homomorphism preserves normality for bDa; and the pre-image of a normal ideal is normal if a is normal. The preliminaries to Theorem 7 state in effect that the operators C, c, and ' preserve complementary C-neutrality for pairs of ideals, yielding by iteration at most three pairs from a given a and a. It follows that normality in our definition is a proper generalization of Stone's in a Boolean algebra. Lemma 12 gives a connection between neutrality and distributivity parallel to that for complementary neutral elements, a, a.