Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction

Details Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction

2004-09-15

arxiv.org/abs/math/0409263v2

Journal of Pure and Applied Algebra 202, no. 1--3 (2005) 201--229

Mathematics

General Mathematics

Scientific paper

A (v,0)-semilattice is ultraboolean, if it is a directed union of finite Boolean (v,0)-semilattices. We prove that every distributive (v,0)-semilattice is a retract of some ultraboolean (v,0)-semilattices. This is established by proving that every finite distributive (v,0)-semilattice is a retract of some finite Boolean (v,0)-semilattice, and this in a functorial way. This result is, in turn, obtained as a particular case of a category-theoretical result that gives sufficient conditions, for a functor $ Pi$, to admit a right inverse. The particular functor $ Pi$ used for the abovementioned result about ultraboolean semilattices has neither a right nor a left adjoint.

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