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# structure

Let $\tau$ be a signature^{}.
A *$\tau$-structure* $\mathcal{A}$ comprises of a set $A$, called the *universe ^{}* (or

*underlying set*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 $n$-ary relation symbol $R\in\tau$, a $n$-ary relation

^{}$R^{A}$ on $A$.

Some authors require that $A$ be non-empty.

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

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

- 1.
A set is a structure, with no constants, no functions, and no relations on it.

- 2.
A partially ordered set is a structure, with one binary relation call partial order defined on the underlying set.

- 3.
A group is a structure, with one binary operation called multiplication, one unary operation called inverse

^{}, and one constant called the multiplicative identity^{}. - 4.
A vector space is a structure, with one binary operation called addition, unary operations called scalar multiplications, one for each element of the underlying set, and one constant $0$, the additive identity

^{}. - 5.
A partially ordered group is a structure like a group, but with the addition of a partial order on the underlying set.

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

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## Comments

## nomenclature

"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?

## another question

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?

## Re: nomenclature

This is standart nomenclature. In some other books or references, the term "model" is also used for structures. The term "model theory" comes from the notion of satisfaction : when we write the symbol A|= phi, we read "A models phi" or "A is a model of phi".

Usually, when the term "model" is used for a "structure", the author means a "structure" that "models" a bunch of formulas. I hope this is clear.

## Re: another question

I think it's just a matter of taste up to some point. It is preferable that models be not empty for fome very technical reasons. The reason why models should be non-empty comes from symtactical concerns, because you would like the completeness theorem to be true.

## Re: nomenclature

So how would you render concepts/sentences like

"countable model of set theory"

"Up to isomorphism, the real numbers are the

unique model of the theory of complete, ordered

fields"

Would your choice of usage be to replace "model"

with "structure" in the above?

## Re: nomenclature

note that you have used "model of a theory" in your question. A "countable model of set theory" is a countable structure which is a model of the axioms of set theory.

Sometimes I've seen "model" used for "structure", especially when we are talking about structures that are models of a theory.

Just to give you a quick example, any graph is a structure for the signature of set theory, but to get a model of set theory, you must have more than a mere graph.

## Re: nomenclature

Thank you. Your point has finally sunk in.

Language -> structure

Theory -> model

Again, thanks for your efforts.