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Flat semilattices


Authors: Sydney Bulman-Fleming and Kenneth McDowell
Journal: Proc. Amer. Math. Soc. 72 (1978), 228-232
MSC: Primary 06A20
DOI: https://doi.org/10.1090/S0002-9939-1978-0505915-X
MathSciNet review: 0505915
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0505915-X
Keywords: Tensor product, distributive semilattice, flat semilattice, killing interpolation property
Article copyright: © Copyright 1978 American Mathematical Society

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