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Classification systems and the decompositions of a lattice into direct products. (English) Zbl 1062.06008
In the paper, the author introduces the notion of a classification system for an arbitrary complete lattice, which has its origin in the formal concept analysis of formal contexts. Namely, a classification system of a complete lattice $$L$$ is a non-empty set $$S=\{\,a_i\,|\,i\in I\,\}$$ of nonzero elements of $$L$$ such that $$a_i\wedge a_j=0$$ for all $$i\neq j$$ and $$x=\bigvee_{i\in I}(x\wedge a_i)$$ for all $$x\in L$$. After listing some basic properties of classification systems, the author presents a characterization of complete pseudocomplemented lattices such that any of their classification systems gives a decomposition of the lattice into a direct product (Theorem 3.7). Finally, these results are used to obtain a description of direct product decompositions of certain pseudocomplemented lattices (Theorem 4.4) and a generalization of G. Grätzer’s and E. T. Schmidt’s characterization of finite distributive Stone lattices as direct products of finite distributive lattices having a least nonzero element (Theorem 4.7).

MSC:
 06B23 Complete lattices, completions 06D15 Pseudocomplemented lattices