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Algebras with one and two direct limits. (English) Zbl 0963.08002
Dorninger, D. (ed.) et al., Contributions to general algebra 12. Proceedings of the 58th workshop on general algebra “58. Arbeitstagung Allgemeine Algebra”, Vienna, Austria, June 3-6, 1999. Klagenfurt: Verlag Johannes Heyn. 211-219 (2000).
Apart from the natural numbers with successor operation, a mono-unary algebra has no proper direct limit if and only if it has no proper retract. For mono-unary algebras each of whose components contains a cycle or the integers with successor operation, the existence of exactly one proper direct limit is equivalent to the existence of exactly one proper retract.
For the entire collection see [Zbl 0942.00022].

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