structure (original) (raw)

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  for each constant symbol c∈τ, anelement cA∈A;
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  for each n-ary function symbol f∈τ, a function (or operation) fA:An→A;
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  for each n-ary relation symbol R∈τ, a n-ary relation RA on A.

Some authors require that A be non-empty.

If 𝒜 is a structure, then the cardinality (or power) of 𝒜, |𝒜|, is the cardinality of its A.

Examples of structures abound in mathematics. Here are some of them:

1. 1. A set is a structure, with no constants, no functions, and no relations on it.
5. 1. A partially ordered group is a structure like a group, but with the addition of a partial order on the underlying set.

If τ contains only relation symbols, then a τ-structure is called a relational structure. If τ contains only function symbols, then a τ-structure is called an algebraic structure. In the examples above, 2 is a relation structure, while 3,4 are algebraic structures.