This book is intended to be a thorough introduction to the subject of
order and lattices, with an emphasis on the latter. It can be used for a
course at the graduate or advanced undergraduate level or for
independent study. Prerequisites are kept to a minimum, but an
introductory course in abstract algebra is highly recommended, since
many of the examples are drawn from this area. This is a book on pure
mathematics: I do not discuss the applications of lattice theory to
physics, computer science or other disciplines. Lattice theory began in
the early 1890s, when Richard Dedekind wanted to know the answer to the
following question: Given three subgroups EF, and G of an abelian group
K, what is the largest number of distinct subgroups that can be formed
using these subgroups and the operations of intersection and sum (join),
as in E?FßÐE?FÑ?GßE?ÐF?GÑ and so on? In lattice-theoretic terms, this is
the number of elements in the relatively free modular lattice on three
generators. Dedekind [15] answered this question (the answer is #))
and wrote two papers on the subject of lattice theory, but then the
subject lay relatively dormant until Garrett Birkhoff, Oystein Ore and
others picked it up in the 1930s. Since then, many noted mathematicians
have contributed to the subject, including Garrett Birkhoff, Richard
Dedekind, Israel Gelfand, George Grätzer, Aleksandr Kurosh, Anatoly
Malcev, Oystein Ore, Gian-Carlo Rota, Alfred Tarski and Johnny von
Neumann.
