Congruence relation

Introduction to Sets and Relations

Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000

ProofFor each x ∈ Z, x − x = 0n. This means that x ≡ x (modulo n) which implies that the congruence relation is reflexive.If x ≡ y (modulo n), x − y = kn for some k ∈ Z. Multiplying both sides of the last equality by −1, we get y − x = −kn which implies that y ≡ x (modulo n). Thus, the congruence relation is symmetrical.If x ≡ y (modulo n) and y ≡ z (modulo n), we have x − y = k1n and y − z = k2n for some k1and k2 in Z. Writing x − z = x − y + y − z, we get x − z = (k1 + k2)n. Since k1 + k2 ∈ Z, we conclude that x ≡ z (modulo n). This shows that the congruence relation is transitive.

Ideals of an EMV-semiring

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Published in International Journal of General Systems, 2020

R. A. Borzooei, M. Shenavaei, A. Di Nola, O. Zahiri

If then .If and for any is a congruence relation with respect to then .

Congruence relation

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Introduction to Sets and Relations

Ideals of an EMV-semiring