Lattice Theory

abstract algebra additive functional Annals of Math attributes automorphism Birkhoff Boolean algebra Borel sets chain condition Chapter closed sets closure complemented modular lattice complete lattice congruence relations contains convex corollary correspondence countable decomposition Dedekind defined definition denote descending chain condition direct union distributive lattice distributive law dual dually isomorphic E. H. Moore elements equivalent example finite dimensions finite-dimensional Hasse diagram Hence homomorphism identity implies isomorphic join Jour L-space lattice of finite lattice theory Lemma linear space matroid measure metric convergence metric lattice modular functionals modular lattice modulo Moreover Neumann normal subspace notion o-complete o)-convergence one-one open sets partially ordered system perspective PG(F points polynomial projective geometry prove representation resp satisfies self-dual semi-modular sense sequence star-convergence subalgebras subgroups sublattice subsets topology transition operators upper bound vector lattice whence x₁ y₁