Compact Semilattices with Open Principal Filters
by
M. Rajagopalan
Tennessee State University, Nashville
Coauthors: Oleg V. Gutik, K. Sundaresan

We study topological semilattices. Locally compact semilattices are of more interest. A topological semigroup is a Hausdorff topological space with a semigroup structure in which multiplication is jointly continuous. A band is a semigroup in which all the elements are idempotents.

A semilattice is a commutative band. Let E be a semilattice. For e, f in E we put e <= f if ef = fe = e, in this case we also write f >= e. We say that e < f if e <= f and e =/= f. If e < f we say that f > e. The relation <= above is called the natural order or the induced order of the semilattice E. An idempotent e of the semilattice E is called maximal (minimal) if f >= e (f <= e) implies f = e for all f in E. An element e of the semilattice E is called a maximum or an identity of E if ef = fe = f for all f in E. (this is equivalent to saying that e >= f for all f in E). If e in E then e is called a minimum or a zero of E if ef = fe = e for all f in E. That is same as saying that e <= f for all f in E. When we use a partial order relation like < , > , <= or >= in E we mean the natural relation in E unless otherwise stated. Further by MaxE (MinE) we mean the set of all maximal elements of E (the set of all minimal elements of E). If e in E then we put \uparrow e = { f | f in E and fe = ef = e}. We put \downarrow e = { f | f in E and fe = ef = f}. If A subset or equal E we put \uparrow A = \cup {\uparrow e | e in A}. We put \downarrow A = \cup {\downarrow e | e in A}. We put NO(e) = E \(\downarrow e \cup \uparrow e).

Definition 1.1 A topological semilattice E is called a semilattice with open principal filters if \uparrow e is open for every e in E.

Definition 1.2 Let E be a topological semilattice. Let e in E. Then e is called a local minimum if there is an open set U containing e so that if ef = fe = f and f in U then f = e. The set of all local minimum in E is denoted as K(E).

Lemma 1.3 A topological semilattice E is a semilattice with open principal filters if and only if E = K(E).

Lemma 1.4 Let E be a topological semilattice with open principal filters. Let U subset or equal E be open. Then \uparrow U is open.

Lemma 1.5 Let E be a topological semilattice. Let e in E. Then \uparrow e and \downarrow e are closed and are also subsemilattices of E. If E is further a semilattice with open principal filters then \uparrow e is both open and closed (that is, clopen).

Lemma 1.6 Let E be a locally compact semilattice with open principal filters. Then E is zero dimensional.

Definition 1.7 A topological lattice which has a basis of open sets which are themselves subsemilattices is called a Lawson semilattice.

Lemma 1.8 A locally compact semilattice with principal ultrafilters is a Lawson semilattice.

Definition 1.9 A topological space X is called scattered if every non-empty closed subset A of X contains a point p that is isolated in A.

Definition 1.10 Let E be a semilattice and A subset or equal E. Let e in A. Then e is called a minimal element of A (a maximal element of A) if whenever f in A and fe = ef = f we have f=e (if whenever f in A and ef = fe = e we have e = f). Also, e is called the largest in A or the maximum in A (the least in A or the minimum in A) if fe = ef = f for all f in A (if fe = ef = e for all f in A).

Lemma 1.11 Every compact nonempty subset A of a topological semilattice E contains a minimal element as well as a maximal element in that subset. If the subset A is also a subsemilattice then it contains a minimum.

Lemma 1.12 A locally compact semilattice E with open principal filters is scattered.

Notation 1.13 If X is a topological space then the set of all its isolated points is denoted by Is(X).

Theorem 1.14 Let E be a locally compact semilattice with open principal filters. Then the following hold:

1. Is(E) is dense in E.
2. w(E) = |E|. (w(E) is weight of E).
3. c(E) = d(E) = |Is(E)| (c(E) is cellularity and d(E) is density character of E).
4. If, in addition, E is compact then \chi(E) = | E |.

Note 1.16 Recall that t(X) denotes the tightness of a topological space. In the above example we have that t( [1, \tau]) = \lambda = |[ 1, \tau] |.

Example 1.17 Let \lambda be an infinite cardinal and X a set of cardinality \lambda with the discerte topology. Let \infty be the only limit point of the one point compactification of X. Put Y = X \cup {\infty}. Define a multiplication in Y by putting xy = \infty if x =/= y and xx = x for all x, y in Y. Then Y is a locally compact semilattice. Notice that |Y| = \lambda and if X is uncountable then t(Y) < |Y|.

Recall that if X is a topological space then \pi_\chi(X) denotes the pseudocharacter of X.

We have the following questions.

Question 1.18 Is there a compact semilattice X so that \pi_\chi(X) < t(X)?

Question 1.19 Does every compact scattered space admit a structure of a compact semilattice with open principal filters?

Question 1.20 Is there a compact semilattice X so that d(X) < \chi(X)?

As far as we know, the last question 1.20 is open. The question 1.19 may not be decidable in ZFC. The question 1.18 is answerable in ZFC. We have the following:

Theorem 1.21 There is a compact semilattice X in ZFC so that \pi_\chi(X) < t(X). In any model of set theory in which the growth \betaN\N of the Stone - Cech compactification \betaN of N has a p-point, there is a compact scattered space which does not admit a structure of a compact semilattice.

Remark 1.22 Ch and MA imply that \betaN\N has p-points. There are models of set theory in which \betaN \N has no p-point.

Recall that a point q of a topological space X is called a p-point if the intersection of a countable number of neighbourhoods of q is again a neighbourhood of q.

Date received: July 17, 2001