On the set of principal congruences in a distributive congruence lattice of an algebra

Czédli, Gábor: On the set of principal congruences in a distributive congruence lattice of an algebra. Acta scientiarum mathematicarum, (84) 3-4. pp. 357-375. (2018)

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Abstract

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 joinreducible 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 result of 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.

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