The endomorphism spectrum of a monounary algebra

Abstract

ABSTRACT. The endomorphism spectrum spec A 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 = spec A. 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 ∈ N it is easy to find a monounary algebra A with {1, 2, . . . , n} = spec A. It will be proved that if i ∈ N, then there exists a monounary algebra A such that spec A 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.

The pre-period of the glued sum of finite modular lattices

“…An endomorphism f of a structure A can be considered as a unary operation and A; f is a monounary algebra. Some properties of monounary algebras connected with the notion of homomorphism were studied, e.g., in [3,4,7,11,12].…”