Note on lattice-ordered groups
Tadasi Nakayama · Proceedings of the Japan Academy Series A Mathematical Sciences · 1942
By a lattice-ordered group, or briefly a lattice-group, we mean a (not necessarily abelian) group which is at the same time a lattice such that the order relation is preserved under left and right multiplica- tion; a b implies ac bc and ca <:__ cb we have ac bc= (a' b)c, ca cb c(a b) too.Abelian lattice-groups , particularly so-called vector lattices , have been studied by many authors.The present short note is to give some simple remarks concerning mainly with non-abelian lattice-groups.We shall begin with elementary observations about homomorphisms.We shall then show that Lorenzen's main theorem for abelian lattice- 2 T. NAKAYAMA.[Vol. 18, vector-lattices with Archimedean units may be obtained also by means of Lorenzen-Clifford's theorem.