Free Banach lattices over pre-ordered Banach spaces exist; the canonical contraction is a bipositive embedding with closed range precisely when the positive wedge is a closed normal cone.
Troitsky,Simple constructions of FBL(A) and FBL[E], Positivity 23 (2019), no
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Free Banach lattices over pre-ordered Banach spaces
Free Banach lattices over pre-ordered Banach spaces exist; the canonical contraction is a bipositive embedding with closed range precisely when the positive wedge is a closed normal cone.