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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 64, Issue 3


The endomorphism spectrum of a monounary algebra

Danica Jakubíková-Studenovská / Katarína Potpinková
Published Online: 2014-07-05 | DOI: https://doi.org/10.2478/s12175-014-0233-7


The endomorphism spectrum specA of an algebra A is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of A, for all endomorphisms of A. In this paper we study finite monounary algebras and their endomorphism spectrum. If a finite set S of positive integers is given, one can look for a monounary algebra A with S = specA. We show that for countably many finite sets S, no such A exists. For some sets S, an appropriate A with spec A = S are described.

For n ∈ ℍ it is easy to find a monounary algebra A with {1, 2, ..., n} = specA. It will be proved that if i ∈ ℍ, then there exists a monounary algebra A such that specA skips i consecutive (consecutive eleven, consecutive odd, respectively) numbers.

Finally, for some types of finite monounary algebras (binary and at least binary trees) A, their spectrum is shown to be complete.

MSC: Primary 08A60; Secondary 08A35

Keywords: finite monounary algebra; endomorphism; endomorphism spectrum; root algebra; (at least) binary tree

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About the article

Published Online: 2014-07-05

Published in Print: 2014-06-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 3, Pages 675–690, ISSN (Online) 1337-2211, DOI: https://doi.org/10.2478/s12175-014-0233-7.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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