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Class preserving mappings of equivalence systems. (English) Zbl 1077.08001

An equivalence system is a pair \(\langle A,\Theta\rangle\) where \(\Theta\) is an equivalence relation on a nonvoid set \(A\). A mapping \(h:A\to B\) is called a class-preserving mapping of systems \(\langle A, \Theta\rangle\) and \(\langle B, \Phi\rangle\) whenever \(h([a]\Theta)= [h(a)]\Phi\), \(a\in A\). It is proved that a class-preserving mapping \(h\) from \(\langle A,\Theta \rangle\) can be characterized by permutability of equivalence relations \(\Theta\) and \(\text{Ker}\,h\). The topic was investigated by H. Werner [Math. Z. 121, 111–140 (1971; Zbl 0203.22902)].

MSC:

08A02 Relational systems, laws of composition
03E20 Other classical set theory (including functions, relations, and set algebra)

Citations:

Zbl 0203.22902
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References:

[1] Madarász R., Crvenković S.: Relacione algebre. : Matematički Institut, Beograd. 1992.
[2] Maltsev A. I.: Algebraic systems. : Nauka, Moskva. 1970)
[3] Riguet J.: Relations binaires, fermetures, correspondances de Galois. Bull. Soc. Math. France 76 (1948), 114-155. · Zbl 0033.00603
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