Flat semilattices

Abstract:

Let S (respectively ${{\mathbf {S}}_0}$) denote the category of all join-semilattices (resp. join-semilattices with 0) with (0-preserving) semilattice homomorphisms. For $A \in {\mathbf {S}}$ let ${A_0}$ represent the object of ${{\mathbf {S}}_0}$ 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 $A \in {\mathbf {S}}\;({{\mathbf {S}}_0})$ is called flat if the functor - $- { \otimes _{\mathbf {S}}}A( - { \otimes _{{{\mathbf {S}}_0}}}A)$ preserves monomorphisms in S $({{\mathbf {S}}_0})$. THEOREM. For $A \in {\mathbf {S}}\;({{\mathbf {S}}_0})$ the following conditions are equivalent: (1) A is flat in S $({{\mathbf {S}}_0})$, (2) ${A_0}(A)$ is distributive (see Grätzer, Lattice theory, p. 117), (3) A is a directed colimit of a system of f.g. free algebras in S $({{\mathbf {S}}_0})$. The equivalence of (1) and (2) in S was previously known to James A. Anderson. $(1) \Leftrightarrow (3)$ is an analogue of Lazard’s well-known result for R-modules.