Retract irreducibility of connected monounary algebras. II.

*(English)*Zbl 0897.08005If \((A,f)\) 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 |

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\textit{D. Jakubíková-Studenovská}, Czech. Math. J. 47, No. 1, 113--126 (1997; Zbl 0897.08005)

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##### References:

[1] | D. Jakubíková-Studenovská: Retract irreducibility of connected monounary algebras I. Czechoslovak Math. J. 46(121) (1996), 291-308. · Zbl 0870.08006 |

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