next up previous
Next: References Up: Clarification to `Congruence Lattices

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:

  equation58
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:

  equation65

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:

equation79

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 tex2html_wrap_inline241 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 tex2html_wrap_inline241 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.

   figure89
Figure 1:

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

Proof. Let tex2html_wrap_inline241 be a compactly generated, semimodular lattice satisfying (1). Let tex2html_wrap_inline263 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 tex2html_wrap_inline241 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 tex2html_wrap_inline241 is independent. Let tex2html_wrap_inline281 be a compact element. Since the atoms join to 1, there is a finite set tex2html_wrap_inline285 of atoms which join above c. Then, by semimodularity and independence,

displaymath205

is a maximal chain in tex2html_wrap_inline289 and so, by (3.8) of [1], every chain is finite. Thus, if we let b be the join of those ai's lying below c and we suppose b < c, then there is an element r with tex2html_wrap_inline301 . By the definition of b, for no ai is tex2html_wrap_inline307 and obviously tex2html_wrap_inline309 and by semimodularity each tex2html_wrap_inline311 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 tex2html_wrap_inline241 is the join of atoms. Let tex2html_wrap_inline317 be the lattice of all subsets of the atoms of tex2html_wrap_inline241 . 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 tex2html_wrap_inline331 then there is an atom  tex2html_wrap_inline333 with tex2html_wrap_inline335 . But since sets of atoms are independent, tex2html_wrap_inline337 implies tex2html_wrap_inline339 ; similarly, tex2html_wrap_inline341 . Thus tex2html_wrap_inline343 , a contradiction. Hence tex2html_wrap_inline241 is isomorphic to the lattice of all subsets of its atoms and thus is certainly distributive. tex2html_wrap_inline278


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 tex2html_wrap_inline241 is a strongly atomic, compactly generated lattice then the following three conditions are equivalent:
  1.   tex2html_wrap_inline241 is locally distributive,
  2.   tex2html_wrap_inline241 is semimodular and every modular sublattice is distributive, and
  3.   for every set of four distinct elements a, p1, p2, tex2html_wrap_inline359 for which p1, p2, tex2html_wrap_inline365 , the sublattice tex2html_wrap_inline367 is an eight-element Boolean algebra.
The first two conditions are equivalent and imply the third, but tex2html_wrap_inline369 shows that the third condition is weaker than the other two. If the third condition is modified by adding the statement that, if p1, tex2html_wrap_inline373 with tex2html_wrap_inline375 , then the sublattice tex2html_wrap_inline377 is a four-element Boolean algebra, the three conditions become equivalent.




next up previous
Next: References Up: Clarification to `Congruence Lattices

Ralph Freese
Wed May 1 14:43:43 HST 1996