Distributivity for Upper Continuous and Strongly Atomic Lattices
Marcin Łazarz, Krzysztof Siemieńczuk · Studia Logica · 2016
In the paper we introduce two conditions (D) and ( $$\hbox {D}^*$$ ) which are strengthenings of Birkhoff’s conditions. We prove that an upper continuous and strongly atomic lattice is distributive if and only if it satisfies (D) and ( $$\hbox {D}^*$$ ). This result extends a theorem of R.P. Dilworth characterizing distributivity in terms of local distributivity and a theorem of M. Ward characterizing distributivity by means of covering diamonds.