Integrally closed and complete ordered quasigroups and loops

Abstract:

We generalize the well-known results on embedding a partially ordered group in its Dedekind extension by showing that, with the appropriate definition of integral closure, any partially ordered quasigroup (loop) G can be embedded in a complete partially ordered quasigroup (loop) if and only if G is integrally closed. If G is directed as well, then its Dedekind extension is a complete lattice-ordered quasigroup (loop). Furthermore, any complete fully ordered quasigroup (loop) has, with one exception, the real numbers with their usual ordering as its underlying set. The quasigroup (loop) operation, however, need not be ordinary addition as it is in the group case. On the other hand, a complete, strongly power associative fully ordered loop is either the integers or the real numbers with ordinary addition.