关于格的极小生成元集

On the Minimum Sets of Generating Elements of a Lattice

本文证明了格的极小生成元集一定是最小生成元集且只能是非零完全并既约元全体，证明了分配格具有最小生成元集的必要条件是它满足并无限分配律．本文还证明了完全Ｈｅｙｔｉｎｇ代数具有最小生成元集当且仅当它是强代数格，证明了完备格是强代数格当且仅当它和它的对偶格均是具有最小生成元集的分配格．

In this paper, we proved that a minimum set of generating elements of a lattice must be the smallest set of generating elements, and it is exactly the set of all completely irreducible elements, proved t hat a necessary condi t io n under wh ich a dist ributive lat t ice possesses the smallest set of generating elements is that it satisfies the sup infinitely distributive law,we also proved that a complete Heyting algebra possesses a minimum set of generating elements if and only it is a strong algebraic lattice, proved that a complete lattice is a strong algebraic lattice if and only if it and its dual lattice are distributive lattices with the smallest set of generating elements.