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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 \(\mathbf L\mathcal A=\mathbf L \mathcal B\cup \left [\mathcal A\right ]\).
08A35 Automorphisms and endomorphisms of algebraic structures
08A60 Unary algebras
08B25 Products, amalgamated products, and other kinds of limits and colimits
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