PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 72, Number 2, November 1978 FLAT SEMILATTICES1 SYDNEY BULMAN-FLEMING AND KENNETH MC DOWELL Abstract. Let S (respectively So) denote the category of all join-semilattices (resp. join-semilattices with 0) with (0-preserving) semilattice homomorphisms. For A G S let A0 represent the object of S0 obtained by adjoining a new 0-element. In either category the tensor product of two objects may be constructed in such a manner that the tensor product functor is left adjoint to the hom functor. An object A eS (Sq) is called flat if the functor - ®gA (- Œ^) preserves monomorphisms in S (So). Theorem. For A 6 S (S0) the following conditions are equivalent: (1) A is fiat in S (Srj), (2) A0 (A) is distributive {see Grätzer, Lattice theory,p. 117), (3) A is a directed colimit of a system of f.g. free algebras in S (So). The equivalence of (1) and (2) in S was previously known to James A. Anderson. (1) «=> (3) is an analogue of Lazard's well-known result for Ä-modules. 1. Introduction. For any algebras A and B in a variety T the tensor product A <S> B can be constructed, and has the defining property that any bi- homomorphism from A x B to an algebra in T factors uniquely through A <8> B. An algebra A E T is called flat if the functor - ®A preserves monomorphisms of T. This terminology is consistent with that used for Ä-modules, and appears also in [1], [4] and [12], while the notion of flatness appearing in [23] is somewhat stronger. In this paper we are concerned with the varieties S (of (V-) semilattices) and S0 (of semilattices with least element 0, henceforth called 0-semilattices). S and S0 will also be considered as categories, the morphisms being all homomorphisms and all 0-preserving homomorphisms, respectively. Tensor products in S and S0, as well as in semigroups, commutative semigroups, distributive lattices and A/-sets, have been extensively studied in recent years. Some idea of the existing literature in this area is given by the references at the end of this paper. In 1969 Lazard [21] proved that an Ä-module M is flat iff it is the directed colimit of a system of finitely generated free Ä-modules. If we call an algebra in a variety °V L-flat if it is the directed colimit of a system of finitely generated T-free algebras, then Lazard's theorem simply states that an Ä-module is L-flat iff it is flat. L-flatness has been studied for Af-sets (M a monoid) in [23], for commutative semigroups in [15], and for arbitrary varieties in [22]. Examples show that the analogue of Lazard's theorem is not true in general: [4] shows that there are M-sets (taking M to be the free Received by the editors October 25, 1977. AMS (MOS) subject classifications (1970). Primary 06A20; Secondary 20M99. Key words and phrases. Tensor product, distributive semilattice, flat semilattice, killing interpo- lation property. 'This research was supported by NRC grants A4494 and A9241. © American Mathematical Society 1978 228 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use