Representations of lattices via neutral elements (Q1064329)

For any neutral element a in a bounded lattice L, the mapping \(x\to (x\bigwedge a,x\bigvee a)\) represents L as a subdirect product of [0,a]\(\times [a,1]\). It is first shown that for certain neutral elements, the image of L under this mapping is completely determined by a homomorphism of [0,a] into [a,1]. Iterating this process, a representation of L can be obtained as a subdirect product of the intervals \([a_ i,a_{i+1}]\) for any chain \(0=a_ 0<a_ 1...<a_ n<a_{n+1}=1\) where each \(a_ i\) is such a neutral element relative to \([0,a_{i+1}]\). The image in this case is completely determined by a family of homomorphisms \(\pi_{i,j}: A_ i\to A_ j\) \((i<j)\), where \(A_ i=[a_ i,a_{i+1}]\).