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Different similarities. (English) Zbl 1373.03049
Summary: We establish the hierarchy among twelve equivalence relations (similarities) on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language $$L$$ and a cardinal $$\kappa$$, we consider the interplay between the restrictions of these similarities to the class $$\mathrm{Mod}_{L}(\kappa)$$ of all $$L$$-structures of size $$\kappa$$. It turns out that, concerning the number of different similarities and the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages (where all these similarities are different).

MSC:
 03C07 Basic properties of first-order languages and structures 03E40 Other aspects of forcing and Boolean-valued models 20M20 Semigroups of transformations, relations, partitions, etc. 06A06 Partial orders, general
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References:
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