Abstract
The following analog of the characterization of flat modules has been obtained for the variety of semimodules over a semiring R: A semimodule R A is flat (i.e., the tensor product functor − ⊗ A preserves all finite limits) iff A is L-flat (i.e., A is a filtered colimit of finitely generated free semimodules). We also give new (homological) characterizations of Boolean algebras and complete Boolean algebras within the classes of distributive lattices and Boolean algebras, respectively, which solve two problems left open in [14]. It is also shown that, in contrast with the case of modules over rings, in general for semimodules over semirings the notions of flatness and mono-.atness (i.e., the tensor product functor − ⊗ A preserves monomorphisms) are different.
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