Retract irreducibility of connected monounary algebras I. (English) Zbl 0870.08006

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 complete characterization of retracts of monounary algebras is given. A connected monounary algebra is referred to as retract irreducible whenever 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 \((A_i,f_i).\) A connected monounary algebra \((A,f)\) possessing a one element cycle \(\{c\}\) is retract irreducible if and only if for any elements \(a\), \(b\) in \(A\) the condition \(f(a)=f(b)\) implies either \(a=b\) or \(c\in \{a,b\}.\)
Reviewer: M.Novotný (Brno)


08A60 Unary algebras


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