The present paper comprises three rather independent notes. The first and second notes are about finite distributive lattices. We hasten to add that the title is a bit unprecise; the third note is actually about finite meet semidistributive lattices. In Section 1 an example of a self-dual distributive lattice, which does not allow for a polarity is given. This answers a question of Kamara. Another question of Metropolis, Rota, and Stein is settled as well. Both problems fit well into the… Expand

It is shown that a large class of self-dual lattices may be endowed with an IRL structure, and examples of lattices which fail to admit IRLs with natural algebraic conditions are given.Expand

A finite analogue of the Birkhoff variety theorem is proved: a non-void class of finite algebras of a finite type τ is closed under the formation of finite products, subalgebras and homomorphic… Expand

An element in a lattice is join-irreducible if x=a∨b implies x=a or x=b. A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every… Expand

It is known that every sublattice A of a free lattice satisfies the following conditions: (W) For all a, b, c, d ∈ A, if ab ≤ c + d, then ab ≤ c or ab ≤ d or a ≤ c+d or b ≤ c + d. (SD) For all u, a,… Expand

ID projective geometry the d u a l i t i e s play an e s s e n t i a l r o l e . The; oan often be desoribed by semibilinear forms of vector spaces coordinatizing geometries. These c l a s s i c a l… Expand

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully… Expand