# zbMATH — the first resource for mathematics

Retract irreducibility of connected monounary algebras. II. (English) Zbl 0897.08005
If $$(A,f)$$ is a monounary algebra, a nonempty subset $$M$$ of $$A$$ is said to be a retract of $$(A,f)$$ whenever there exists an endomorphism $$h$$ of $$(A,f)$$ onto $$(M,f)$$ such that $$h(x)=x$$ for any $$x$$ in $$M.$$ A connected monounary algebra is referred to as retract irreducible if it has the following property: If it is isomorphic to a retract of a product $$\prod _{i\in I}(A_i,f_i)$$ of some connected monounary algebras, then it is isomorphic to a retract of some factor $$(A_i,f_i).$$ In Part I [Czech. Math. J. 46, No. 2, 291-308 (1996; Zbl 0870.08006)], the author found all retract irreducible connected monounary algebras possessing a one-element cycle. In the present paper the following theorem is proved. A connected monounary algebra which has no one-element cycle is retract irreducible if and only if either it is a cycle with $$p^n$$ elements where $$p$$ is a prime and $$n$$ a natural number or it is isomorphic to the monounary algebra of all natural numbers with the successor operation.
Reviewer: M.Novotný (Brno)

##### MSC:
 08A60 Unary algebras
##### Keywords:
connected monounary algebra; retract irreducibility
Full Text:
##### References:
 [1] D. Jakubíková-Studenovská: Retract irreducibility of connected monounary algebras I. Czechoslovak Math. J. 46(121) (1996), 291-308. · Zbl 0870.08006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.