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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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