Flat semilattices

Abstract

Let S (respectively S 0 {{\mathbf {S}}_0} ) denote the category of all join-semilattices (resp. join-semilattices with 0) with (0-preserving) semilattice homomorphisms. For A ∈ S A \in {\mathbf {S}} let A 0 {A_0} represent the object of S 0 {{\mathbf {S}}_0} 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 ∈ S ( S 0 ) A \in {\mathbf {S}}\;({{\mathbf {S}}_0}) is called flat if the functor - − ⊗ S A ( − ⊗ S 0 A ) - { \otimes _{\mathbf {S}}}A( - { \otimes _{{{\mathbf {S}}_0}}}A) preserves monomorphisms in S ( S 0 ) ({{\mathbf {S}}_0}) . THEOREM. For A ∈ S ( S 0 ) A \in {\mathbf {S}}\;({{\mathbf {S}}_0}) the following conditions are equivalent: (1) A is flat in S ( S 0 ) ({{\mathbf {S}}_0}) , (2) A 0 ( A ) {A_0}(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 ( S 0 ) ({{\mathbf {S}}_0}) . The equivalence of (1) and (2) in S was previously known to James A. Anderson. ( 1 ) ⇔ ( 3 ) (1) \Leftrightarrow (3) is an analogue of Lazard’s well-known result for R-modules.