## 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)

### Keywords:

equivalence relation

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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