SemigroupForum, Vol. 7 (1974), 372-374. A REMARK ON LATTICE-ORDERED SEMIGROUPS Laszlo Fuchs To Alfred H. Clifford on his 65th birthday on the llth of July, 1973 Several relevant results have been obtained concern- ing the structure of fully ordered semigroups (one of the pioneers being Alfred H. Clifford), but not much is known of the structure of lattice-ordered semigroups in general. A natural way to begin with is to study those lattice- ordered semigroups which are subdirect products of fully ordered semigroups. These form a subvariety in the vari- ety of all lattice-ordered semigroups (namely, the sub- variety generated by the fully ordered semigroups; of. Fuchs [I, Thm. 5]), thus it is of primary importance to know the laws which characterize this subvariety. The general case seems to be rather complicated: we do not even know if there is a finite basis for the laws. In the commutative case, however, it is fairly easy to show that some of the obviously necessary laws characterize the subvariety in question. This easy special case is settled in this short note. For a class of lattice-ordered semirings, an analo- gous result has been established by Smith [2]. All semigroups under consideration are commutative and lattice-ordered. For simplicity, we assume that all contain an identity e which is at the same time the maximum element. The variety C of all these semigroups is thus defined in terms of a 0-ary operation: e, and three binary operations: ., u, n, satisfying the 372 © 1974bySpringer-VerlagNew Yorklnc.