Factoring a lattice
Alexander Doniphan Wallace · Proceedings of the American Mathematical Society · 1958
For the purpose of this note a lattice is a Hasdorff space L together with a pair A, V. LXL>-*L of continuous functions satisfying the usual conditions [3, p. 18]. The closed unit interval I with the operations min, max is a simple example. An iseomorphism is a simultaneous isomorphism and homeomorphism (A. H. Clifford). It is known and easy to prove [1] that a closed and bounded interval admits a unique (modulo an iseomorphism or a dual iseomorphism) structure. It is known [1] that a 2-cell admits more than one lattice structure. We shall show that if no maximal chain cuts L into more than two pieces then the lattice structure is unique. If L1 and L2 are lattices then L1XL2 is the cartesian product of LI and L2 with coordinate operations.