The endomorphism spectrum of a monounary algebra

Rok, strany: 2014, 675 - 690

The endomorphism spectrum $spec\mathcal{A}$ of an algebra $\mathcal{A}$ is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of $\mathcal{A}$, for all endomorphisms of $\mathcal{A}$. 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 $\mathcal{A}$ with $S=spec\mathcal{A}$. We show that for countably many finite sets $S$, no such $\mathcal{A}$ exists. For some sets $S$, an appropriate $\mathcal{A}$ with $spec\mathcal{A}=S$ are described. For $n\in\mathbb{N}$ it is easy to find a monounary algebra $\mathcal{A}$ with $\{1,2,...,n\}=spec\mathcal{A}$. It will be proved that if $i\in\mathbb{N}$, then there exists a monounary algebra $\mathcal{A}$ such that $spec\mathcal{A}$ skips $i$ consecutive (consecutive eleven, consecutive odd, respectively) numbers. Finally, for some types of finite monounary algebras (binary and at least binary trees) $\mathcal{A}$, their spectrum is shown to be complete.

ISO 690:

Jakubíková-Studenovská, D., Potpinková , K. 2014. The endomorphism spectrum of a monounary algebra. In Mathematica Slovaca, vol. 64, no.3, pp. 675-690. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0233-7

APA:

Jakubíková-Studenovská, D., Potpinková , K. (2014). The endomorphism spectrum of a monounary algebra. Mathematica Slovaca, 64(3), 675-690. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0233-7