The minimal closed monoids for the Galois connection ${\rm End}$-${\rm Con}$
Danica Jakubíková-Studenovská, Reinhard Pöschel, Sándor Radeleczki
Received October 3, 2022. Published online June 14, 2023.
Abstract: The minimal nontrivial endomorphism monoids $M={\rm End}{\rm Con} (A,F)$ of congruence lattices of algebras $(A,F)$ defined on a finite set $A$ are described. They correspond (via the Galois connection ${\rm End}$-${\rm Con}$) to the maximal nontrivial congruence lattices ${\rm Con} (A,F)$ investigated and characterized by the authors in previous papers. Analogous results are provided for endomorphism monoids of quasiorder lattices ${\rm Quord}(A,F)$.
References: [1] E. Halušková: Strong endomorphism kernel property for monounary algebras. Math. Bohem. 143 (2018), 161-171. DOI 10.21136/MB.2017.0056-16 | MR 3831484 | Zbl 1463.08003
[2] E. Halušková: Some monounary algebras with EKP. Math. Bohem. 145 (2020), 401-414. DOI 10.21136/MB.2019.0128-18 | MR 4221842 | Zbl 1499.08011
[3] D. Jakubíková-Studenovská: On congruence relations of monounary algebras I. Czech. Math. J. 32 (1982), 437-459. DOI 10.21136/CMJ.1982.101820 | MR 0669786 | Zbl 0509.08003
[4] D. Jakubíková-Studenovská: On congruence relations of monounary algebras II. Czech. Math. J. 33 (1983), 448-466. DOI 10.21136/CMJ.1983.101895 | MR 0718928 | Zbl 0535.08003
[5] D. Jakubíková-Studenovská, J. Pócs: Monounary Algebras. P. J. Šafárik University, Košice (2009). Zbl 1181.08001
[6] D. Jakubíková-Studenovská, R. Pöschel, S. Radeleczki: The lattice of quasiorder lattices of algebras on a finite set. Algebra Univers. 75 (2016), 197-220. DOI 10.1007/s00012-016-0373-4 | MR 3515397 | Zbl 1338.08005
[7] D. Jakubíková-Studenovská, R. Pöschel, S. Radeleczki: The lattice of congruence lattices of algebras on a finite set. Algebra Univers. 79 (2018), Article ID 4, 23 pages. DOI 10.1007/s00012-018-0486-z | MR 3770896 | Zbl 1414.08001
[8] D. Jakubíková-Studenovská, R. Pöschel, S. Radeleczki: The structure of the maximal congruence lattices of algebras on a finite set. J. Mult.-Val. Log. Soft Comput. 36 (2021), 299-320. Zbl 07536105
[9] L. Janičková: Monounary algebras containing subalgebras with meet-irreducible congruence lattice. Algebra Univers. 83 (2022), Article ID 36, 10 pages. DOI 10.1007/s00012-022-00786-1 | MR 4462594 | Zbl 07573924
[10] H. Länger, R. Pöschel: Relational systems with trivial endomorphisms and polymorphisms. J. Pure Appl. Algebra 32 (1984), 129-142. DOI 10.1016/0022-4049(84)90048-3 | MR 0741962 | Zbl 0558.08004
[11] P. P. Pálfy: Unary polynomials in algebras. I. Algebra Univers. 18 (1984), 262-273. DOI 10.1007/BF01203365 | MR 0745492 | Zbl 0546.08005
[12] R. Quackenbush, B. Wolk: Strong representation of congruence lattices. Algebra Univers. 1 (1971), 165-166. DOI 10.1007/BF02944974 | MR 0295980 | Zbl 0231.06006
Affiliations: Danica Jakubíková-Studenovská, Institute of Mathematics, Faculty of Science, P. J. Šafárik University, Šrobárova 2, 041 80 Košice, Slovakia, e-mail: Danica.Studenovska@upjs.sk; Reinhard Pöschel (corresponding author), Institute of Algebra, Faculty of Mathematics, Technische Universität Dresden, 01069 Dresden, Germany, e-mail: reinhard.poeschel@tu-dresden.de; Sándor Radeleczki, Institute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary, e-mail: sandor.radeleczki@uni-miskolc.hu
