## 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)

### MSC:

 08A60 Unary algebras

### Keywords:

unary algebras; monounary algebra; retract irreducibility

### References:

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