分子格的既约度,分解度与分子格乘积的既约性

Degree of irreducibility and degree of decomposition for a molecular lattice and irreducibility of product of molecular lattices

为探讨分子格的乘积的既约性和乘积的既约分解,文献[1]提出了分子格的既约度和分解度的概念,本文是[1]的继续,进一步给出了分子格的既约度与分解度的一些性质,证明了关于分子格乘积的全体主子格之集的基数的一个定理,以及得出了一族分子格的乘积是既约分子格的一个充要条件,此外,本文还证明了分子格范畴中乘积对上积的完全分配性是自然同构,改正了[1]中对这一结论的证明。

Concepts of degree of irreducibility and degree of decomposition for a molecular lattice were put forward in [1], this paper is a continuation of [1]. In this paper, we give properties of degree of irreducibility and degree of decomposition for a molecular lattice, prove a theorem on the cardinal number of the set of principal sublattices of the product of a family of molecular lattices, and obtain a necessary and sufficient condition under which the product of a family of molecular lattices is an irreducible molecular lattice. Besides, we show that the isomorphism of the complete distributivity of the product operation for the co—operation in the category of molecular lattices is a natural isomorphism, and correct the proof in [1] for this result.