On strong semimodular lattices. III (Q1118625)

It is well-known that an arbitrary complemented modular lattice is relatively complemented. For the case of finite length this result was extended by the author [On complemented strong lattices, Prepr. Sekt. Math., Martin-Luther-Univ. Halle Wittenberg 103 (1987)] from modular lattices to strong semimodular lattices. Moreover it could be shown that in this case it is sufficient to require instead of complementedness that only each atom of the lattice has a complement. In this paper we show that this and other results follow easily from two inequalities which were derived by \textit{G. Richter} and the author [Strong elements in lattices of finite length, ibid. 106 (1987)]. This paper is a loose continuation of Parts I and II [the author, ibid. 99 and 102 (1987)], where other properties of strong semimodular lattices were investigated.