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congruence (Definition)

Let $ \Sigma$ be a fixed signature, and $ \mathfrak{A}$ a structure for $ \Sigma$. A congruence $ \sim$ on $ \mathfrak{A}$ is an equivalence relation such that for every natural number $ n$ and $ n$-ary function symbol $ F$ of $ \Sigma$, if $ a_i \sim a_i'$ then $ F^\mathfrak{A}(a_1, \ldots a_n) \sim F^\mathfrak{A}(a_1', \ldots a_n').$



"congruence" is owned by almann.
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See Also: congruence relation on an algebraic system
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Cross-references: function symbol, natural number, equivalence relation, structure, signature, fixed
There are 8 references to this entry.

This is version 6 of congruence, born on 2003-07-20, modified 2003-08-02.
Object id is 4486, canonical name is Congruence3.
Accessed 2323 times total.

Classification:
AMS MSC: 03C05 (Mathematical logic and foundations :: Model theory :: Equational classes, universal algebra)
 03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)
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