By Steffen Koenig and Changchang Xi
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Proof. If a is a pseudo-complement, of b say, in L and a < c is essential then b 1\ c 0 and thus a 1\ (b 1\ c) "I o. But a 1\ (b 1\ c) = (a 1\ b) 1\ c = 0, a contradiction. 4. But c is a pseudo-complement of b in L, hence a is a c pseudo-complement. FINITENESS CONDITIONS FOR LATTICES. Remark. 4. (up to going over to 1(L) in order to apply the above to the upper-continuous case). 4. follows from this elegant argument. e. if and only if [O,a] is simple. An upper-continuous lattice with and 1 is semiatomic if 1 is a join of atoms.
Consider a =f. 0 in L and let b be a pseudo-complement of a. Clearly b =f. I and thus [b,l] contains an atom, c say. Since b < c it follows that a /\ c =f. O. Put d = a /\ c and consider 0 < x :::; d. Since x :::; a and x :::; c we deduce that b < b V x :::; c. Since c is an atom in [~,l] it follows that b V x = c, and hence d /\ (x V b) = x V (d /\ b) = x V (a /\ c /\ b) = x V (a /\ b /\ c) = x V 0 = x. Therefore d /\ c = x or d = x and it follows that d is an atom. From d :::; a it follows that a contains an atom.
If M is an R-module, resp. a finitely generated R-module, the there exists a free module, resp. a finitely generated module, L say, together with a submodule K of L such that L(M) ~ [K,L]. Proof. 8. and apply the above. FINITENESS CONDITIONS FOR MODULES. If the family (Ni}iEI is join-independent then we say that the family is an independent family of submodules of Mj in this case we will have that EiEI Ni ~ iE1 Ni -+ EiEI Ni, (Xi}iEI f-+ EiEI Xi, where iE1 Ni · Indeed, define f: Xi E Ni . Clearly, f is surjective.
On the structure of cellular algebras by Steffen Koenig and Changchang Xi