Represent the three element chain as a subdirect product of subdirectly irreducible lattices.

Question

Represent the three element chain as a subdirect product of subdirectly irreducible lattices. I know the two element chain as either a Boolean algebra or a semilattice,is subdirectly irreducible.In fact , a distributive lattice subdirectly irreducible if and only if it has exactly two elements. Any finite chain with two or more elements,as a Heyting algebra,is subdirectly irreducible.(This is not case for chain of three or more elements as either lattices or semilattices,which are subdirectly reducible to the two-element chain.) Now I want given an example for the three element chain as a subdirect product of subdirectly irreducible lattices. please help me. thanks.

Answer 1

The 3-element chain $C$ has just two non-trivial congruences, namely one that identifies the middle element with the top and one that identifies the middle element with the bottom. The projections of $C$ into the two quotients combine to give you an embedding of $C$ into the (4-element) product of the two quotients, and the image of that embedding is the subdirect product you want.