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From: afzal@cui.unige.ch (Afzal Ballim)
Newsgroups: sci.math,comp.theory
Subject: Posets and Lattices
Keywords: Lattices, Posets
Message-ID: <453@cui.unige.ch>
Date: 14 Nov 89 11:29:09 GMT
Organization: University of Geneva, Switzerland
Lines: 42

Hi there,
    I have a problem relating to lattices. I am currently reading "Lattice
Theory" by T. Donnellan. On page 53 of this book, there is a theorem that a
finite lattice has a zero (or least) element, and a unity (or greatest)
element. The theorem itself is simple, based on properties of meet and join.
However, on page 55, at the end of the theorem there is the following
paragraph:

"This theorem indicates how we may form a lattice of n elements from a
partially ordered set of n-2 elements by adding a zero element o and a unity
element u;" [Donnellan, pp.  55].

My problem is this: surely he could not be saying what he seems to be
saying, i.e., that any partial order of n-2 elements can be made into a
lattice of n elements by adding a zero and a unity element. This is
blatantly false. The simplest poset that contradicts this is the abstract
poset of 4 elements:
                   H = {a,b,c,d}
with the following partial ordering:
	  {c <= a, c <= b, d <= a, d <= b} 
The addition of a zero and a unity element to H does not yield a lattice,
because the meet of a and b (a v b) is undefined (the lower bounds of a and
b have no greatest element) and the join of c and d (c ^ d) is also
undefined (their upper bounds have no least element).

So, am I missing something? Is the statement in Donnellan wrong? Or, is
there something missing? For example, must it be the case that the partial
ordering on the set must allow one to define an algebraic operation over the
set for the conclusion (above) to apply?

Any help on this would be appreciated.

-Afzal Ballim
 afzal@divsun.unige.ch
 Institut Dalle Molle pour les Etudes Semantiques et Cognitives
 University of Geneva, Switzerland



References:

Donnellan, T. (1968) "Lattice Theory."  Pergamon Press: Oxford.