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From: moss@takahe.cs.umass.edu (Eliot &)
Newsgroups: sci.math,comp.theory
Subject: Re: Posets and Lattices
Message-ID: <MOSS.89Nov15084428@takahe.cs.umass.edu>
Date: 15 Nov 89 13:44:28 GMT
Article-I.D.: takahe.MOSS.89Nov15084428
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In-reply-to: afzal@cui.unige.ch's message of 14 Nov 89 11:29:09 GMT

I will admit to not being an expert in this area, though I have studied
lattices and posets a bit for understanding how they are used in some styles
of defining the semantics of programming languages. The point is that the two
added elements (conventionally called "bottom" and "top" and drawn as an
upside down T and a regular T in the papers with which I am familiar) are
*defined* to be the meet/join of elements that do not otherwise have a meet or
join. That is what makes the thing a lattice. The Hasse diagram for the poset
you gave is this:

	a   b
        |\ /|
        | X |
        |/ \|
        c   d

The Hasse diagram for the lattice adding 0 and 1 as bottom and top
respectively is this:

	   1
	  / \
	 a   b
	 |\ /|
	 | X |
	 |/ \|
	 c   d
	  \ /
	   0

This is, I believe, a perfectly good lattice. I suppose you did not take into
account that in adding the two new elements to the poset, one also adds <=
relationships as well. In particular, for all elements x, 0 <= x <= 1. I hope
this solves your difficulties ....
--

		J. Eliot B. Moss, Assistant Professor
		Department of Computer and Information Science
		Lederle Graduate Research Center
		University of Massachusetts
		Amherst, MA  01003
		(413) 545-4206; Moss@cs.umass.edu