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Normally presented varieties. (English) Zbl 0842.08007
An identity $$p(x_1,\dots, x_n) = q(x_1,\dots, x_n)$$ is called normal if it is of the form $$x_1 = x_1$$ or neither $$p$$ nor $$q$$ is a variable. For a variety $$\mathcal V$$, $${\mathcal N} ({\mathcal V})$$ denotes the variety defined by all normal identities of $$\mathcal V$$. Let $${\mathfrak A} \in {\mathcal V}$$, $$\Theta$$ any congruence on $$\mathfrak A$$ and $$\kappa$$ any function sending each $$\Theta$$ class to an element of the class and satisfying some non-triviality property. The author defines the choice algebras: For any $$n$$-ary term $$f(x_1, \dots, x_n)$$ and for elements $$a_1, \dots, a_n$$ let $$f_{(\Theta,\kappa)} (a_1, \dots, a_n) = \kappa([f(a_1, \dots, a_n)]\Theta)$$. The author proves that $${\mathcal N}({\mathcal V})$$ consists of the homomorphic images of the choice algebras of the given variety. Some related questions and examples are discussed.
Reviewer: E.Fried (Budapest)

##### MSC:
 08B99 Varieties
##### Keywords:
normal identities; choice algebras
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##### References:
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