Re: three-element lattice expression and Polya's enumeration

No, Marc, you haven't misunderstood, your answer is precisely what I was
looking for, thanks!

But I do not quite understand your other remark. If you enumerate all
possible combinations of binary relations between different pairs of
elements from a finite set of n elements of a lattice, implying that these
elements of course already satisfy all the relations that lattice elements
are required to, and then go on to show that all these combinations share a
particular property, then you have shown the property to hold for n-tuples
of elements of any lattice, haven't you? You point out, in fact, prove, that
such enumeration is not necessary for proving the general validity of the
given example. But Davey & Priestley in their Introduction to Lattices &
Order seem to suggest that enumeration of cases is sometimes necessary for
proving [nonmodularity] or [modularity but nondistributivity] (p. 92). Or
are they (I am still studying their text)?

Peter van Emburg,
Leiden.


"Marc Olschok" <sa796ol@xxxxxxxxxxxxxxxxxxxxxx> wrote in message
news:3ngod1F1dlojU1@xxxxxxxxxxxxxxxxx
> Peter van Emburg <petervanemburg@xxxxxxxxx> wrote:
>> L.S.,
>>
>> am I going about it the wrong way or is it indeed necessary to enumerate
>> all
>> distinct results of selecting with replacement three elements of a
>> lattice
>> for proving a lemma about lattices such as
>>
>> a /\ (b \/ c) >= (a /\ b) \/ (a /\ c) ?
>
> At least in the above example, it is not necessary.
> Just recall the definition of \/ and /\ via
>
> (I) x <= u /\ v <==> x <= u and x <= v
> (S) u \/ v <= x <==> u <= x and v <= x
>
>
> From (a /\ b) <= a and (a /\ b) <= b <= (b \/ c)
> follows (via (I))
>
> (1) (a /\ b) <= a /\ (b \/ c)
>
> From (a /\ c) <= a and (a /\ c) <= c <= (b \/ c)
> follows (via (I))
>
> (2) (a /\ c) <= a /\ (b \/ c)
>
> From (1) and (2) follows (via (S))
>
> (a /\ b) \/ (a /\ c) <= a /\ (b \/ c)
>
> In general, I doubt if enumeration helps a lot in trying to show
> that some results holds for _every_ lattice. Of course I might
> have misunderstood your original question.
>
> Marc


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