# Characterizing fully principal congruence representable distributive lattices

- 23 Downloads

**Part of the following topical collections:**

## Abstract

Motivated by a recent paper of G. Grätzer, a finite distributive lattice *D* is called *fully principal congruence representable* if for every subset *Q* of *D* containing 0, 1, and the set *J*(*D*) of nonzero join-irreducible elements of *D*, there exists a finite lattice *L* and an isomorphism from the congruence lattice of *L* onto *D* such that *Q* corresponds to the set of principal congruences of *L* under this isomorphism. A separate paper of the present author contains a necessary condition of full principal congruence representability: *D* should be planar with at most one join-reducible coatom. Here we prove that this condition is sufficient. Furthermore, even the automorphism group of *L* can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Grätzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.

## Keywords

Distributive lattice Principal lattice congruence Congruence lattice Principal congruence representable Simultaneous representation Automorphism group## Mathematics Subject Classification

06B10## References

- 1.Baranskiĭ, V.A.: On the independence of the automorphism group and the congruence lattice for lattices. In: Abstracts of Lectures of the 15th All-Soviet Algebraic Conference, vol. 1, Krasnojarsk, p. 11 (1979)Google Scholar
- 2.Czédli, G.: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices. Algebra Univ.
**67**, 313–345 (2012)MathSciNetCrossRefMATHGoogle Scholar - 3.Czédli, G.: Representing a monotone map by principal lattice congruences. Acta Math. Hung.
**147**, 12–18 (2015)MathSciNetCrossRefMATHGoogle Scholar - 4.Czédli, G.: The ordered set of principal congruences of a countable lattice. Algebra Univ.
**75**, 351–380 (2016)MathSciNetCrossRefMATHGoogle Scholar - 5.Czédli, G.: An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices. Acta Sci. Math. (Szeged)
**82**, 3–18 (2016)MathSciNetCrossRefMATHGoogle Scholar - 6.Czédli, G.: Representing some families of monotone maps by principal lattice congruences. Algebra Univ.
**77**, 51–77 (2017)MathSciNetCrossRefMATHGoogle Scholar - 7.Czédli, G.: Cometic functors and representing order-preserving maps by principal lattice congruences. Algebra Univ. (submitted)Google Scholar
- 8.Czédli, G.: On the set of principal congruences in a distributive congruence lattice of an algebra. Acta Sci. Math. (Szeged) (to appear)Google Scholar
- 9.Czédli, G., Grätzer, G.: Planar semimodular lattices and their diagrams. In: Grätzer, G., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications, pp. 91–130. Birkhäuser, Basel (2014)Google Scholar
- 10.Czédli, G., Lenkehegyi, A.: On classes of ordered algebras and quasiorder distributivity. Acta Sci. Math. (Szeged)
**46**, 41–54 (1983)MathSciNetMATHGoogle Scholar - 11.Freese, R.: The structure of modular lattices of width four with applications to varieties of lattices. Mem. Am. Math. Soc.
**9**(181), vii+91 (1977)MathSciNetMATHGoogle Scholar - 12.Grätzer, G.: The order of principal congruences of a bounded lattice. Algebra Univ.
**70**, 95–105 (2013)MathSciNetCrossRefMATHGoogle Scholar - 13.Grätzer, G.: Congruences and prime-perspectivities in finite lattices. Algebra Univ.
**74**, 351–359 (2015)MathSciNetCrossRefMATHGoogle Scholar - 14.Grätzer, G.: Homomorphisms and principal congruences of bounded lattices I. Isotone maps of principal congruences. Acta Sci. Math. (Szeged)
**82**, 353–360 (2016)MathSciNetCrossRefMATHGoogle Scholar - 15.Grätzer, G.: Characterizing representability by principal congruences for finite distributive lattices with a join-irreducible unit element. Acta Sci. Math. (Szeged)
**83**, 415–431 (2017). https://doi.org/10.14232/actasm-017-036-7 - 16.Grätzer, G.: Homomorphisms and principal congruences of bounded lattices. II. Sketching the proof for sublattices. Algebra Univ.
**78**(3), 291–295 (2017). https://doi.org/10.1007/s00012-017-0461-0 - 17.Grätzer, G.: Homomorphisms and principal congruences of bounded lattices. III. The independence theorem. Algebra Univ.
**78**(3), 297–301 (2017). https://doi.org/10.1007/s00012-017-0462-z - 18.Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. I. Construction. Acta Sci. Math. (Szeged)
**73**, 445–462 (2007)MathSciNetMATHGoogle Scholar - 19.Grätzer, G., Lakser, H.: Notes on the set of principal congruences of a finite lattice. I. Some preliminary results. Algebra Univ. (submitted, available from ResearchGate and https://arxiv.org/abs/1705.05319)
- 20.Grätzer, G., Quackenbush, R.W.: Positive universal classes in locally finite varieties. Algebra Univ.
**64**, 1–13 (2010)MathSciNetCrossRefMATHGoogle Scholar - 21.Grätzer, G., Schmidt, E.T.: The strong independence theorem for automorphism groups and congruence lattices of finite lattices. Beitr. Algebra Geom.
**36**, 97–108 (1995)MathSciNetMATHGoogle Scholar - 22.Grätzer, G., Sichler, J.: On the endomorphism semigroup (and category) of bounded lattices. Pac. J. Math.
**35**, 639–647 (1970)MathSciNetCrossRefMATHGoogle Scholar - 23.Grätzer, G., Wehrung, F.: The strong independence theorem for automorphism groups and congruence lattices of arbitrary lattices. Adv. Appl. Math.
**24**, 181–221 (2000)MathSciNetCrossRefMATHGoogle Scholar - 24.Urquhart, A.: A topological representation theory for lattices. Algebra Univ.
**8**, 45–58 (1978)MathSciNetCrossRefMATHGoogle Scholar