×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] GRÄTZER G.: Universal Algebra. The University Series in Higher Mathematics, D. Van Nostrand, Co., Princeton, N.Y., 1968. · Zbl 0182.34201
[2] HALUŠKOVÁ E.: On iterated direct limits of a monounary algebra. Contributions to General Algebra 10, Heyn, Klagenfurt, 1998, pp. 189-195. · Zbl 0912.08003
[3] HALUŠKOVÁ E.: Direct limits of monounary algebras. Czechoslovak Math. J. 49 (1999), 645-656. · Zbl 1004.08003
[4] HALUŠKOVÁ E.-PLOŠČICA M.: On direct limits of finite algebras. Contributions to General Algebra 11, Heyn, Klagenfurt, 1999, pp. 101-104. · Zbl 0931.08005
[5] HALUŠKOVÁ E.: Monounary algebras with two direct limits. Math. Bohem. 125 (2000), 485-495. · Zbl 0966.08005
[6] HALUŠKOVÁ E.: Algebras with one and two direct limits. Contributions to General Algebra 12, Heyn, Klagenfurt, 2000, pp. 211-219. · Zbl 0963.08002
[7] JAKUBÍK J.-PRINGEROVÁ G.: Direct limits of cyclically ordered groups. Czechoslovak Math. J. 44 (1994), 231-250. · Zbl 0821.06015
[8] JÓNSSON B.: Topics in Universal Algebra. Lecture Notes in Math. 250, Springer Verlag, Berlin, 1972. · Zbl 0225.08001
[9] NOVOTNÝ M.: Über Abbildungen von Mengen. Pacific J. Math. 13 (1963), 1359-1369. · Zbl 0137.25304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.