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On direct limit classes of algebras. (English) Zbl 1150.08001
Let $$S$$ be the class of all algebras $$\mathcal A$$ such that every surjective or injective endomorphism of $$\mathcal A$$ is an automorphism. The system of all isomorphic copies of a class $$\mathcal K$$ of algebras is denoted by $$\left [\mathcal K\right ]$$. Further, for an algebra $$\mathcal A$$, the system of all algebras which are endomorphic images of $$\mathcal A$$ is denoted by $$\mathbf E \mathcal A$$ and the system of all algebras which are isomorphic to some algebra obtained by direct limits from $$\mathcal A$$ is denoted by $$\mathbf L \mathcal A$$. In the present paper, classes of algebras of the form $$\mathbf L \mathcal A$$ are investigated, generalizing some results from monounary algebras. The main result of the paper is the following theorem: Let $$\mathcal A,\mathcal B\in S$$ be such that $$\left [\mathbf E\mathcal A\right ]=\left [\mathcal A,\mathcal B\right ]$$ and $$\mathbf E\mathcal B=\{\mathcal B\}$$. Then $$\mathbf L\mathcal A=\left [\mathcal A,\mathcal B\right ]$$. As a corollary it is shown that if $$\mathcal A\in S$$ and $$\mathcal B$$ is a subalgebra of $$\mathcal A$$ with $$\mathbf E \mathcal A=\mathbf E \mathcal B\cup \{\mathcal A\}$$, $$\mathbf L \mathcal B\subseteq\mathbf L \mathcal A$$, then $$\mathbf L\mathcal A=\mathbf L \mathcal B\cup \left [\mathcal A\right ]$$.
##### MSC:
 08A35 Automorphisms and endomorphisms of algebraic structures 08A60 Unary algebras 08B25 Products, amalgamated products, and other kinds of limits and colimits
##### Keywords:
direct limit; endomorphism; monounary algebra
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##### References:
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