The endomorphism spectrum of a monounary algebra. (Q2877064)

Finite monounary algebras and their endomorphism spectra are studied. The notion of an endomorphism spectrum of an algebraic structure was introduced by \textit{K. Grant}, \textit{R. J. Nowakovski} and \textit{I. Rival} [Order 12, No. 1, 45-55 (1995; Zbl 0849.06002)]. The endomorphism spectrum \(\mathrm{spec\,}\mathcal A\) 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 \(\mathcal A\).NEWLINENEWLINE All two-element sets \(S\) such that there exists a monounary algebra \(\mathcal A\) with \(\mathrm{spec\,}\mathcal A=S\) are determined. Let \(i\) be a positive integer. Monounary algebras \(\mathcal A\) such that \(\mathrm{spec\,}\mathcal A\) skip \(i\) consecutive numbers, \(i\) consecutive even numbers and \(i\) consecutive odd numbers are constructed. It is proved that if \(\mathcal A\) is so called at least binary tree, then \(\mathrm{spec\,}\mathcal A =\{1,2,\ldots,|A|\}\).