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I found the following definition for a structure in my math course:

A structure $X$ consists of a non-empty set $D_x$, the universum of $X$ and the attribution of values $r^x$ to non logical-symbols $r$:

-The value $c^x$ for an object-symbol $c$ is an element of the universum $D_x$. $c$ can be a constant value or a variable.

-The value $F^x$ for a function-symbol $F/n$ is a function $F^x:D_x^n \rightarrow D_x$. This is a function that maps n-values $(a_1,...,a_n)$ from the universum to single values of the universum.

-The value $P^x$ for a predicate-symbol $P/n$ is a n-relation $P^x$ in $D_x$, so $P^x \subset D_x^n$.

We call $r^x$ the value or interpretation of $r$ in X.

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This seems pretty abstract to me, Can someone give me an example of a structure with this definition? Or just more information/explanation? I looked on the internet but didn't find anything.

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Ayoub Rossi
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1 Answers1

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You can see Peano arithmetic and the corresponding structure of natural numbers:

$(\mathbb N, 0, S, +, \times)$.

Here $\mathbb N = \{ 0,1,2,\ldots \}$ is the domain $D$ and $0$ is the only individual constant $c$ denoting the number zero.

$S(x)$ is a function symbol that is interpreted with the successor function, i.e. $\text {Succ} : \mathbb N \to \mathbb N$.

Finally, $+$ and $\times$ are binary function symbols, interpreted with sum and product respectively.

If we want to consider also an example of relation, we have to consider the binary predicate symbol $<$, that will be interpreted with the "less then" relation, i.e. $\text {less} \subseteq \mathbb N \times \mathbb N$.

Mauro ALLEGRANZA
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  • Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really? – Ayoub Rossi Jan 23 '19 at 16:19
  • @AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, \ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,\times$ and $<$ all have arity $2$ (they are binary functions/relations). – Mauro ALLEGRANZA Jan 23 '19 at 16:25
  • Is a mathematical object an item in such a structure? That is can you say every mathematical object belongs to a structure? – Erik Engheim Apr 20 '20 at 11:43
  • @ErikEngheim - a mathematical structure is a mathematical object made of mathematical objects... $\mathbb N$ is "made of" natural numbers with their properties and operations. – Mauro ALLEGRANZA Apr 20 '20 at 11:50
  • @MauroALLEGRANZA thanks! so mathematical object is a more abstract/generic concept? So a natural number is a mathematical object only, while ℕ is a mathematical object AND a structure? – Erik Engheim Apr 20 '20 at 12:12