Projective and injective distributive lattices
Raymond Balbes · Pacific Journal of Mathematics · 1967
This paper is concerned with the properties of projective and injective distributive lattices.By considering the minimal Boolean extension of a distributive lattice L, the question of the injectivity of L is transferred to the category of Boolean algebras, where a characterization is known.The result is that L is injectivein the category of distributive lattices-if and only if it is a complete Boolean algebra.The first section deals with a method of defining E-fΐβe sequences of elements in a distributive lattice.Roughly speaking, these are elements which satisfy a given set E of inequalities and no others except consequences of E.We prove that a finite distributive lattice is projective if and only if the sum of any two meet irreducible elements is meed irreducible.For the general case we show that a distributive lattice is projective if and only if it is generated by an E-fΐee sequence, where E is a certain set of one-sided inequalities.The last section concerns the projectivity of Boolean algebras, chains, and direct products.