PlanetMath: structure

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structure

(Definition)

Let $\tau$ be a signature. A $\tau$ -structure $\mathcal{A}$ comprises of a set , called the universe (or domain) of $\mathcal{A}$ , and an interpretation of the symbols of $\tau$ as follows:

for each constant symbol $c\in\tau$ , an element $c^A\in A$ ;
for each -ary function symbol $f\in\tau$ , a function $f^A:A^n\rightarrow A$ ;
for each -ary relation symbol $R\in\tau$ , a subset $R^A\subseteq A^n$ .

If $\mathcal{A}$ is a structure, then the cardinality (or power) of $\mathcal{A}$ , $\vert\mathcal{A}\vert$ , is the cardinality of its universe .

"structure" is owned by CWoo. [ full author list (3) | owner history (3) ]

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See Also: relational structure

Also defines:

structure, interpretation

Attachments:: relational structure (Derivation) by CWoo

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Cross-references: power, cardinality, subset, relation symbol, function, function symbol, constant symbol, signature
There are 329 references to this entry.

This is version 15 of structure, born on 2002-06-03, modified 2007-11-16.
Object id is 3017, canonical name is StructuresAndSatisfaction.
Accessed 21270 times total.

Classification:

AMS MSC:

03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)

Pending Errata and Addenda

None.[ View all 4 ]

Discussion

forum policy

another question by rmilson on 2002-06-03 20:03:09

In regards to your structures entry, here is a
question that has been bothering me for a good long while:

Are empty models/structures allowed?

Of course, all foralls are true in an empty model,
and all "there exists" false.

Usually the standard texts say "no empty models allowed"
and I always wondered: "how come?"

The closest I've been able to come to an explanation,
is that people seem to want to be able to deduce
that

(For all x) (Px) |- (Exists x)(Px)

and to do this, you need to forbid empty models.

This always struck me as a question of convention,
an arbitrary decision someone made long ago.

Am I missing something? What is so abhorrent about
empty models that they must be forbidden?

[ reply | up ]

Re: another question by jihemme on 2002-06-03 20:33:09

nomenclature by rmilson on 2002-06-03 19:56:19

"Structures" or "models" that is the question.

I am no logician, but in every encounter I have had
with logic, this concept was named a "model".
Witness the fact that "model" theory is an established branch of
mathematical logic.

So my question is: where is your "structure" terminology
coming from. Is this a personal preference, or
is there a pattern of widespread usage to back up
your choice of words?

[ reply | up ]

Re: nomenclature by jihemme on 2002-06-03 20:26:00
- Re: nomenclature by rmilson on 2002-06-04 11:03:36
  - Re: nomenclature by jihemme on 2002-06-04 11:13:50
    - Re: nomenclature by rmilson on 2002-06-04 11:45:51

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