Lattice question

1. Dec 14, 2003

phoenixthoth

by lattice, i mean definition 2 at http://en2.wikipedia.org/wiki/Complete+lattice [Broken] .

this is probably ill-posed but here goes nothing...

is there a nonempty lattice such that the maximal element of the lattice is the whole lattice?

so what i mean by a maximal element is an element b such that for all a in the lattice, a <= b. is there a lattice such that such a b is the whole lattice?

Last edited by a moderator: May 1, 2017
2. Dec 14, 2003

Hurkyl

Staff Emeritus
I'm not sure what you mean by that; how can an element of the lattice be the lattice?

3. Dec 14, 2003

HallsofIvy

The only way I can think to do that is if the lattice consists of a single element and you identify the lattice with that element.

4. Dec 14, 2003

phoenixthoth

i'm thinking some kind of self-similarity would be involved. perhaps that a lattice of only one element is the only possible answer. thanks.

5. Dec 15, 2003

phoenixthoth

lattice set theory

i'm sure this has been tried before but perhaps one can approach set theory through lattices as defined in definition 2 of http://en2.wikipedia.org/wiki/Complete+lattice [Broken].

two axioms of lattice set theory would be
1. there is an element in the lattice, Ø, such that for all x,
x^Ø=Ø and x v Ø=x and
2. there is an element in the lattice, U, such that for all x,
x^U=x and x v U=U.

another would be that x^y=y^x and x v y=y v x.

i'm suspecting there might be a problem when one allows x to be U or Ø in the two axioms:
1. (x=U). U^Ø=Ø and U v Ø = U.
2. (x=Ø). Ø^U=Ø and Ø v U = U. ok, i guess there's no contradiction so far.

i'm trying to avoid fuzzy logic, if possible, at least for right now.

one of the main issues is how to restate a version of the subsets axiom. i think the definiion of subset would have to be that x is a subset of y if and only if x<=y which means x v y=y. i'd like to have a subsests axiom so that given a y and well-formed-formula (wff) p, there is an x such that z &isin; x iff (z &isin; y and p(z)). just a thought: in two-valued logic, p(z) is either true or false. maybe i can switch and to meet, ^, and define p(z) to be U if p(z) is true and Ø if p(z) is false. the other problem will be to define &isin; . i'd want it to be defined in terms of meet and join and so that x is a subset of y if and only if (z &isin; x implies z &isin; y). one random candidate is that x &isin; y would be the same as x<y which means that (x!=y and x<=y). well, whatever &isin; means, S(y,p):={z &isin; y : p(z)} could be defined so that z &isin; S(y,p) iff (z &isin; y)^p(z) or something...

however i handle the subsets axiom, i want to avoid russell's paradox, of course. that would be the case of s:=S(U,p) where p(z) says z ! &isin; z when one asks if s &isin; s.

Last edited by a moderator: May 1, 2017