The free modular lattice, ${\rm FM}(2+2+2)$, is infinite

Abstract

The following are known concerning the cardinality of the free modular lattice generated by a set of distinct chains. 1. FM(1+1+1) contains 28 elements [1, p. 68]. 2. FM(2+1+1) contains 138 elements [2, p. 1]; [3, p. 629]. 3. FM(1 + 1 + 1 + 1) contains an infinite chain of distinct elements [4, p. 114]. The proof is due to G. Birkoff. 4. FM(nl+n2), nl, n2 finite, is a finite distributive lattice [1, p. 72]. By a method similar to the one used to prove 3 above, the following theorem is proved. THEOREM. FM(2 + 2 + 2) contains an infinite chain of distinct elements. PROOF. Let G be the free Abelian group (additive) generated by X1, X2, , Xn x c, For k-0, 1, 2, * c . let us define the following. a, [X3k+1 + X3k+2], the subgroup generated by the set of all X3k+1 +X37j+2;