## Lattice Theory |

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abstract Annals of Math ascending chain condition automorphisms Banach Birkhoff Boolean algebra cardinal number cardinal product closed sets closure closure algebra commutative complemented modular lattice complete lattice congruence relations contains Conversely Corollary correspondence countable define definition denote direct union directed set distributive lattice dually Duke Jour elements equivalent ergodic Example finite length functions G. D. Birkhoff hence homomorphic implies infinite integers intersection interval isomorphic isotone lattice of finite Lemma linear logic m-lattice matroid metric lattice modular lattice modular law Moreover non-void one-one open sets order topology ordered group ordinal partly ordered set permutable po-group points prime ideal Problem projective geometry prove real numbers relatively complemented result ring semi-modular lattice semigroup sequence Show subalgebra subgroups sublattice subset subspaces Theorem theory topology transition operators upper bound valuation vector lattice whence Z-group Z-ideals