# `Expanding' one lattice by another one

From: Stephan Lehmke (Stephan.Lehmke@cs.uni-dortmund.de)
Date: Thu Mar 22 2001 - 03:39:49 MET

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(Crossposted between comp.ai.fuzzy and sci.math; followup2 set to
sci.math.)

Dear Colleagues,

In [1] I have defined the `expansion' of a lattice by another one
thusly:

Let [L,^,v] and [L',^',v'] be arbitrary complete lattices with 0, 1
and 0', 1'.

Then an L'-fuzzy filter in L is a homomorphism from [L,^,1] into
[L',^',1']. An L'-fuzzy filter F in L is called principal if there
exist a in L and d in L' such that for every b in L,

/ 1', if b = 1
|
F(b) = < d, if not b = 1 and a <= b
|
\ 0', if not a <= b

where <= denotes the partial order induced by the lattice structure on
L (compare [2]).

The usual fuzzy subset relation induces on the set of all L'-fuzzy
filters in L a complete lattice Fl(L',L).

Then a complete lattice L'' is called an expansion of L by L' iff L''
is isomorphic with a sublattice of the dual of Fl(L',L) containing all
fuzzy principal filters.

It is obvious that L and the dual lattice of L' are embeddable into
every expansion of L by L'.

In my PhD thesis, I need expansions as label structures for labelled
fuzzy logics, with interesting results. Now I'm writing the last
chapter and I'd like to know whether this concept has been used
anywhere else ;-)

I know [2] and [3], but they have different aims and don't mention
anything like this.

Maybe something analogous is studied under a different name (the term
"expansion" is my invention)?

Or the name "expansion" is used for another mathematical concept?

Any hints or pointers are gratefully appreciated.

regards and TIA
Stephan

[1]
@INPROCEEDINGS{Lehmke97b,
language = "USenglish",
booktitle = "FUZZ-IEEE '97 --- Sixth IEEE International Conference on Fuzzy Systems",
month = jul # { 1--5,},
year = 1997,
author = {Stephan Lehmke},
title = "Some Properties of Fuzzy Ideals on a Lattice",
volume = "II",
pages = "813-818"
}

[2]
@ARTICLE{Gaehler95a,
language = "USenglish",
author = {Werner G"ahler},
title = "The general fuzzy filter approach to fuzzy topology, {I}",
journal = "Fuzzy Sets and Systems",
year = 1995,
volume = "76",
pages = "205-224"}

[3]
@ARTICLE{Yuan/Wu90,
language = "USenglish",
author = "Bo Yuan and Wangming Wu",
title = "Fuzzy Ideals on a Distributive Lattice",
journal = "Fuzzy Sets and Systems",
year = 1990,
volume = 35,
pages = "231-240"}

```--
Stephan Lehmke     		 Stephan.Lehmke@cs.uni-dortmund.de
Fachbereich Informatik, LS I	 Tel. +49 231 755 6434
Universitaet Dortmund		 FAX 		  6555
D-44221 Dortmund, Germany
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