Clarification to `Congruence Lattices of Semilattice'

Ralph Freese and J. B. Nation

May 1, 1996

This note contains some clarifications for our paper Congruence lattices of semilattices, [2]. In that paper we use the term relatively pseudo-complemented for the condition:

  i.e., each principal filter is pseudo-complemented. Based on the way relatively complemented lattices are defined (every interval is complemented), this seemed like the natural way to define relatively pseudo-complemented (for a lattice with a greatest element, (1) is equivalent to each each interval being pseudo-complemented). However, the term relatively pseudo-complemented was already in use in algebraic logic. There it was defined as:

 

Unfortunately, (1) and (2) are not the same; in fact, (2) implies distributivity. However, (1) and (2) are equivalent for distributive lattices. Thus (1) could have been used to define relatively pseudo-complemented without changing the meaning for distributive lattices.

Of course, our statement (3) of [2] that the congruence lattice, Con(L), of a semilattice S, is relatively pseudo-complemented means that it satisfies (1). It is worth pointing out that for compactly generated lattice condition (1) is equivalent to meet semidistributivity:

In statement (4) of [2] we claim that Con(L) is locally distributive. This is true, but our argument stated that a compactly generated lattice is locally distributive if and only if it is semimodular and satisfies (1). We referred to Crawley and Dilworth's book [1]. However, Crawley and Dilworth prove this under the assumption that is strongly atomic. The lattice diagrammed in Figure 1 is compactly generated and locally distributive but is not semimodular. Thus one direction of the above equivalence does not hold without strong atomicity. However, the following theorem shows that the other direction does hold and thus our statement (4) of [2] is correct.

  
Figure 1:

Theorem 1. If a compactly generated, semimodular lattice satisfies (1) then it is locally distributive.

Proof. Let be a compactly generated, semimodular lattice satisfying (1). Let and let u be the join of the covers of z. We need to show the interval sublattice u/z is distributive. Since every interval of also is compactly generated, semimodular and satisfies (1), we may assume z=0 and u=1, i.e., 1 is the join of the atoms.

Note that (1) implies that any set of atoms of is independent. Let be a compact element. Since the atoms join to 1, there is a finite set of atoms which join above c. Then, by semimodularity and independence,

is a maximal chain in and so, by (3.8) of [1], every chain is finite. Thus, if we let b be the join of those a_i's lying below c and we suppose b < c, then there is an element r with . By the definition of b, for no a_i is and obviously and by semimodularity each either equals or covers b. But it is easy to see that this violates (1).

We conclude that every compact element, and thus every element, of is the join of atoms. Let be the lattice of all subsets of the atoms of . Map the elements of B to L by mapping each subset to its join in L. By what we have just shown, this map is onto and it clearly preserves joins. Let B and C be sets of atoms. If then there is an atom  with . But since sets of atoms are independent, implies ; similarly, . Thus , a contradiction. Hence is isomorphic to the lattice of all subsets of its atoms and thus is certainly distributive.

Most of the above arguments are in Crawley and Dilworth's book. We will take this opportunity to point out a small error in that book. On page 53 they state that if is a strongly atomic, compactly generated lattice then the following three conditions are equivalent:

1.   is locally distributive,
2.   is semimodular and every modular sublattice is distributive, and
3.   for every set of four distinct elements a, p₁, p₂, for which p₁, p₂, , the sublattice is an eight-element Boolean algebra.

The first two conditions are equivalent and imply the third, but shows that the third condition is weaker than the other two. If the third condition is modified by adding the statement that, if p₁, with , then the sublattice is a four-element Boolean algebra, the three conditions become equivalent.

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