Lattice-ordered loops and quasigroups

AbstractIn studying the effect of an order on non-associative systems such as loops or quasigroups, a natural question to ask is whether some order condition which implies commutativity in the group case implies associativity in the corresponding loop case. For example, a well-known theorem (Birkhoff, [1]) concerning lattice ordered groups states that if the descending chain condition holds for the positive elements, then the l.o. group is actually a direct product of infinite cyclic groups with its partial order induced in the usual way by the linear order in the factors. It is easy to show (Zelinski, [6]) that a fully-ordered loop satisfying the descending chain condition on positive elements is actually an infinite cyclic group. In this paper we generalize this result and Birkhoff's result, by showing that any lattice-ordered loop with descending chain condition on its positive elements is associative. Hence, any l.o. loop with d.c.c. on its positive elements is a free abelian group. More generally, any lattice-ordered quasigroup in which bounded chains are finite, is isotopic to a free abelian group. These results solve a problem in Birkhoff's Lattice Theory, 3rd ed. The proof uses only elementary properties of loops and lattices