Notes on sectionally complemented lattices. II Generalizing the 1960 sectional complement with an application to congruence restrictions

摘要

G. Gratzer and H. Lakser proved in 1986 that for the finite distributive lattices D and E, with |D| > 1, and for the {0, 1}-homomorphism φ of D into E, there exists a finite lattice L and an ideal I of L such that D ≡ Con L, E ≡ Con I, and φ is represented by the restriction map. In their recent survey of finite congruence lattices, G. Gratzer and E. T. Schmidt ask whether this result can be improved by requiring that L be sectionally complemented. In this note, we provide an affirmative answer. The key to the solution is to generalize the 1960 sectional complement (see Part I) from finite orders to finite preorders.