Representations of lattices via neutral elements

Abstract

For any neutral element a in a bounded latticeL, the mapping x→(x∧,x∨a) representsL as a subdirect product of [0, a]×[a, 1]. It is first shown that for certain neutral elements, the image ofL under this mapping is completely determined by a homomorphism of [0, a] into [a, 1]. Iterating this process, a representation ofL can be obtained as a subdirect product of the intervals [ai, ai+1] for any chain 0=a0<a1... <an<an+1=1 where each ai is such a neutral element relative to [0, ai+1]. The image in this case is completely determined by a family of homomorphisms πi,j:Ai →Aj(i<j), where Ai=[ai, ai+1].