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Direct limits of monounary algebras. (English) Zbl 1004.08003
Let \((P,\leq)\) be a nonempty directed partially ordered set, \(A_p\) a monounary algebra for any \(p \in P\). Suppose that the carriers of the algebras \(A_p\) are mutually disjoint and that for any pair of elements \(p\) and \(q\) in \(P\) with \(p<q\), a homomorphism \(\varphi _{pq}\) of \(A_p\) into \(A_q\) is defined such that \(p<q<s\) implies that \(\varphi _{ps} = \varphi _{pq} \circ \varphi _{qs}\). For any \(p\in P\) let \(\varphi _{pp}\) be the identity on \(A_p\). Then \(\{A_p\}_{p\in P}\) is said to be a directed family of monounary algebras. If \(p,q\in P\) and \(x\in A_p\), \(y\in A_q\), then \(x \equiv y\) means that there exists \(s\in P\) with \(p\leq s\), \(q\leq s\), \(\varphi _{ps} (x) = \varphi _{qs} (y)\). Then \(\equiv \) is a congruence on the monounary algebra \(\bigcup _{p\in P} A_p\) and the factor algebra \(\bigcup _{p\in P} A_p/ \equiv \) is called the direct limit of the directed family \(\{A_p\}_{p\in P}\) (cf. G. Grätzer [Universal algebra. Van Nostrand, Princeton (1968; Zbl 0182.34201)]). A class of monounary algebras is said to be a direct limit class if its closure with respect to isomorphisms is closed with respect to direct limits.
The author proves that some classes of monounary algebras are direct limit classes, e.g.: the class of all monounary algebras, the class of all connected monounary algebras, the class of all connected monounary algebras having a cycle with \(k\) elements (\(k\geq 1\) an integer), the class of all cycles with \(k\) elements (\(k \geq 1\) an integer), the class of all monounary algebras having either no cycles or one-element cycles, the class of all monounary algebras consisting of cycles such that the number of elements in a cycle in the algebra does not divide the number of elements in another cycle of the algebra (class \(T\)).
A monounary algebra \(A\) is said to have the property \((C)\) whenever the following holds: If an algebra \(B\) can be obtained as a direct limit of algebras which are isomorphic to \(A\), then \(B\) is isomorphic to \(A\). A monounary algebra \(A\) has property \((C)\) if and only if one of the following cases occurs: (a) \(A\) is a member of \(T\); (b) \(A\) is isomorphic to the set of all integers (with the operation of succession). Furthermore, any direct limit class contains an algebra with property \((C)\).
Remark. The corollaries numbered with 5, 6, 7 should be renumbered with 1, 2, 3.

MSC:
08A60 Unary algebras
08B25 Products, amalgamated products, and other kinds of limits and colimits
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