# Non-separable center of a lattice (Solved)

 Importance: Medium ✭✭
 Author(s): Porton, Victor
 Subject: Algebra
 Keywords: lattice
 Posted by: porton on: September 11th, 2008
 Solved by:

I will call center of a bounded distributive lattice the sublattice of all complemented elements of .

I will call a bounded distributive lattice a lattice with separable center when

Equivalently a bounded distributive lattice with separable center is such a bounded distributive lattice that

Conjecture   There exist bounded distributive lattices which are not with separable center.

(Previously this problem was erroneously stated without the word distributive, it is corrected now.)

This conjecture follows from William Elliot's post in sci.math and the criterion of a lattice to be distributive, see here.

So the problem solves positively.

Admins: Should I delete this problem from Open Problem Garden because its solution was too easy (found in less than one day). Or just to leave it marked as solved?

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