| 1. | This equivalence relation is a semigroup congruence, as defined above.
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| 2. | The affine concept of parallelism forms an equivalence relation on lines.
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| 3. | Binary relations that are both reflexive and Euclidean are equivalence relations.
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| 4. | This defines an equivalence relation on the set of almost homomorphisms.
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| 5. | Since x \ sim _ j y is an equivalence relation.
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| 6. | For example, an equivalence relation possesses cycles but is transitive.
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| 7. | Any homomorphism defines an equivalence relation on by if and only if.
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| 8. | Like any equivalence relation, a semigroup congruence \ sim induces congruence classes
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| 9. | In the next step, one imposes a set of equivalence relations.
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| 10. | This equivalence relation is an abstraction of the germ equivalence described above.
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