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.
['Marcin Łazarz', 'Krzysztof Siemieńczuk']
Distributivity for Upper Continuous and Strongly Atomic Lattices