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On clones determined by their initial segments. (English) Zbl 1163.08001
Let $$\mathcal A$$ and $$\mathcal B$$ be abstract clones. We say that they are locally isomorphic if their $$n$$-segments are isomorphic for every $$n\in\omega$$. It is proven for bounded finitary similarity type that two algebras of this type have isomorphic polynomial clones if and only if they have locally isomorphic ones. Examples of algebras of the same type such that 5mm
a)
their type is unbounded finitary and their polynomial clones are locally isomorphic but not isomorphic,
b)
they have one binary and countably many unary operations and their term clones are locally isomorphic but not isomorphic,
c)
they have two unary operations and their centralizer clones are locally isomorphic but not isomorphic
are presented. For a class $$\mathcal C$$ of clones, properties of their representability as centralizer clones in categories are investigated.

##### MSC:
 08A40 Operations and polynomials in algebraic structures, primal algebras 03C60 Model-theoretic algebra 08B25 Products, amalgamated products, and other kinds of limits and colimits 18B15 Embedding theorems, universal categories
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