## Lattice ordered groups |

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a a b a-th level a,b e Archimedean group cardinal sum Clearly completely distributive convex subgroup COROLLARY define disjoint set element in G essential subgroups exists finite basis finite number finite valued following are equivalent free abelian group free group free set g e G G is Archimedean G is finite G is representable G that contains G the following group G homomorphism ideal of G integers intersection isomorphism Jl-group lattice order least upper bound Lemma Let G lex-extension lex-sum maximal component minimal plenary subset minimal prime subgroup normal subgroup o-isomorphism polar Proof PROPOSITION regular subgroup right o-group right order root system satisfies F satisfies the DCC special elements structure theorem subdirect sum subgroup of G subsemigroup subset of G subspace t-ideal Theorem totally ordered set ultrafilter v-hull value of g x e G y e G