Geodesic
Construction of a complementary quasiorder
Algebra and discrete mathematics, Tome 25 (2018) no. 1, pp. 39-55.

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For a monounary algebra $\mathcal{A}=(A,f)$ we study the lattice $\operatorname{Quord}\mathcal{A}$ of all quasiorders of $\mathcal{A}$, i.e., of all reflexive and transitive relations compatible with $f$. Monounary algebras $(A, f)$ whose lattices of quasiorders are complemented were characterized in 2011 as follows: ($*$) $f(x)$ is a cyclic element for all $x \in A$, and all cycles have the same square-free number $n$ of elements. Sufficiency of the condition ($*$) was proved by means of transfinite induction. Now we will describe a construction of a complement to a given quasiorder of $(A, f)$ satisfying ($*$).
Keywords: monounary algebra, lattice, complemented lattice.
Mots-clés : quasiorder, complement 
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Danica Jakubíková-Studenovská; Lucia Janičková. Construction of a complementary quasiorder. Algebra and discrete mathematics, Tome 25 (2018) no. 1, pp. 39-55. http://geodesic.mathdoc.fr/item/ADM_2018_25_1_a4/

[1] J. Berman, “On the congruence lattice of unary algebras”, Proc. Amer. Math. Soc., 36 (1972), 34–38 | DOI | MR | Zbl

[2] D. P. Egorova and L. A. Skornyakov, “On the congruence lattice of unary algebras”, Uporyad. Mnozhestva Reshetki, 4 (1977), 28–40 (Russian) | MR | Zbl

[3] D. Jakubíková-Studenovská, “Lattice of quasiorders of monounary algebras”, Miskolc Math. Notes, 10 (2009), 41–48 | DOI | MR | Zbl

[4] D. Jakubíková-Studenovská and M. Petrejčíková, “Complemented quasiorder lattices of monounary algebras”, Algebra Universalis, 65 (2011), 363–369 | DOI | MR | Zbl