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Extra resources for Semimodular Lattices

Example text

In accordance with our previous notation we denote the meat of all dual atoms of L by 1+. If 1+ > o, then there exists an atom p(E L) such that p S 1+. It follows that all dual atoms of L are comparable with p, that is, no dual atom can be a complement of p. 3 (s. KALMAN ). If the congruence lattice of an algebra is atomic and if the atoms have complements, then this algebra is isomorphic to a subdirect product of simple algebras. 4. Consider an atomic algebraic lattice. If the meet of all its dual atoms is 0, then each atom has a complement.

Rss xa[r,s] rss xa[r,s] It remains to show that L ~Q(r,s) = 1 for x E J~k(L). rss xa[r,s] For fixed x E J~k(L) we set K d~f {[r,s] E Int(Q):xa[r,s]}. By (ii),K forms an order ideal of (Int(Q),•J. Let (ri' si] be the maximal elements of K for 1 SiS m. Setting Ki d'f ([ri' si]) we have K ~ K1 v ••• v Km. By (iii) there exists atE Q with {x} U a(x) • Lt. We show that [t,t] E K for 1 S i S m. With the help of (i) we conclude Lri n Lsi n Lt s LriAt n Lsivt and therefore k S la(x) n (Lri n Lsi)l = = la(x) n CLri n Lsi n Lt)l S la(x) n LriAt n Lsivtl Thus [riA t, si v t] belongs to K and by the maximality of [ri,s] it follows that ri S t S si and hence [t,t] E Ki.

E. g. PEZZOLI ). This means, in particular, that in a finite distributive lattice L we have, among others, the equalities (++) (+) IJ(L)i = IM(L)i and l(M(L)) = l(J(L)), where 1 denotes the length of the corresponding posets. Some attention has been paid to the question of establishing similar connections between J(L) and M(L) for a finite modular lattice L and, more generally, for a finite semimodular lattice L. Although the arithmetical theory of modular lattices, let alone semimodular lattices, is far more intricated than the arithmetic of distributive lattices, there are some positive results in this direc- 31 tion.