On inverse limits of monounary algebras

Abstract

ABSTRACT. We study inverse limits of monounary algebras. All monounary algebras A such that A can arise from A only by an inverse limit construction are described. We deal with an existence of an inverse limit. Some inverse limit closed classes are described. The paper ends with two problems. Direct and inverse limits are well-known methods for building up new algebras from given families of algebras. The construction of an inverse limit is dual to the construction of a direct limit, which yields that they have a lot of common properties, cf., e.g., [1: §21]. The analogous notions are applied also in the theory of categories under names directed colimit and directed limit, cf., e.g. [12].The paper [5] is devoted to direct limits. In the present paper we investigate inverse limits of monounary algebras. Direct limits of monounary algebras were studied in [2].It is possible that an inverse limit of an inverse family does not exist. Examples of such families are given in Section 3. In Section 2 we will prove that there are proper classes K such that if an inverse family consists of algebras of K, then this inverse family has a limit.Theorem 1 describes all monounary algebras A such that {A} is an inverse limit closed class, i.e. such that they satisfy the following condition:If an algebra B can be obtained as an inverse limit of algebras which are isomorophic to A, then B is isomorphic to A.The class of unbounded monounary algebras from [9] is important in the proof. Another inverse limit closed classes are described in Theorems 2 and 3. All of them are proper classes.2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 08A60, 08B25; Secondary 03C05. K e y w o r d s: monounary algebra, inverse limit, closed class, term operation, retract. This work was supported by grants VEGA 2/0028/13 and VEGA 2/0035/11. Unauthenticated Download Date | 5/10/18 5:25 AM

The endomorphism spectrum of a monounary algebra

“…A rather large series of further algebraic notions is based on endomorphisms. Some properties of monounary algebras connected with the notion of homomorphism were studied by several authors ( [5], [10], [3], [11], [12]). …”