Flat Semilattices

Abstract

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 (-OE^) 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.