2017/05/30 by Czédli, Gábor
#06B10 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1705.10833
Let Q be a subset of a finite distributive lattice D. An algebra A represents the inclusion Q⊆ D by principal congruences if the congruence lattice of A is isomorphic to D and the ordered set of principal congruences of A corresponds to Q under this isomorphism. If there is such an algebra for every subset Q containing 0, 1, and all join-irreducible elements of D, then D is said to be fully (A1)-representable. We prove that every fully (A1)-representable finite distributive lattice is planar and it has at most one join-reducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is fully chain-representable in the sense of a recent paper of G. Grätzer. Combining the results of this paper with another paper by the present author, it follows that every fully (A1)-representable finite distributive lattice is "fully representable" even by principal congruences of finite lattices. Finally, we prove that every chain-representable inclusion Q⊆ D can be represented by the principal congruences of a finite (and quite small) algebra.