Abstract
A homomorphism quasi-order between algebraic structures \(\mathcal {A}\) and \({\mathcal{B}}\) of the same type is defined as follows: \({\mathscr{A}}\leq {\mathscr{B}}\) if there is a homomorphism of \({\mathscr{A}}\) to \({\mathscr{B}}\). The paper deals with the class \({\mathcal{L}}\) of all connected monounary algebras. Let \(\sim \) be the equivalence corresponding to the relation ≤ and let \(\mathbb {L}={\mathcal{L}}\sim \). We prove that \(\mathbb {L}\) is a lattice. Next it is shown that the lattice \(\mathbb {L}\) is distributive. For the purpose of the proof, to each connected monounary algebra without a cycle, an increasing sequence of ordinals is assigned. Finally, we deal with the special case of increasing sequences, namely, such that the sequence contains only positive integers.
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Jakubíková–Studenovská, D. Homomorphism Order of Connected Monounary Algebras. Order 38, 257–269 (2021). https://doi.org/10.1007/s11083-020-09539-y
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DOI: https://doi.org/10.1007/s11083-020-09539-y