Definition of Substructures According to Wikipedia
I have a quick question about the definition for substructures in mathematics from its Wikipedia page:
What does it mean by functions and relations are "traces" of the functions and relations of the bigger structure? What does traces mean?
logic definition model-theory
asked Nov 29 at 0:08
numericalorange
1,719311
add a comment |
I have a quick question about the definition for substructures in mathematics from its Wikipedia page:
What does it mean by functions and relations are "traces" of the functions and relations of the bigger structure? What does traces mean?
logic definition model-theory
asked Nov 29 at 0:08
numericalorange
1,719311
6
Restrictions. $ $
– Asaf Karagila♦
Nov 29 at 0:10
add a comment |
I have a quick question about the definition for substructures in mathematics from its Wikipedia page:
What does it mean by functions and relations are "traces" of the functions and relations of the bigger structure? What does traces mean?
logic definition model-theory
asked Nov 29 at 0:08
numericalorange
1,719311
I have a quick question about the definition for substructures in mathematics from its Wikipedia page:
What does it mean by functions and relations are "traces" of the functions and relations of the bigger structure? What does traces mean?
logic definition model-theory
logic definition model-theory
asked Nov 29 at 0:08
numericalorange
1,719311
asked Nov 29 at 0:08
numericalorange
1,719311
asked Nov 29 at 0:08
numericalorange
1,719311
asked Nov 29 at 0:08
numericalorange
1,719311
asked Nov 29 at 0:08
numericalorange
1,719311
1,719311
6
Restrictions. $ $
– Asaf Karagila♦
Nov 29 at 0:10
add a comment |
6
Restrictions. $ $
– Asaf Karagila♦
Nov 29 at 0:10
6
6
Restrictions. $ $
– Asaf Karagila♦
Nov 29 at 0:10
Restrictions. $ $
– Asaf Karagila♦
Nov 29 at 0:10
add a comment |
1 Answer
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If $Xsubseteq Y$ and $f:Y^nrightarrow Y$, we can look at how $f$ behaves on $X$: that is, look at the function $$f_X: X^nrightarrow Y.$$ Note that a priori this isn't a function from $X^n$ to $X$: $f$ might "go outside" of $X$, even if we just feed it things from $X$.
This $f_X$ is the restriction of $f$ to $X$ (I've never heard it called the "trace" before, but I wouldn't be too surprised if someone calls it that). If $ran(f_X)subseteq X$, then $(X;f_X$) is indeed a structure, and "lives inside" $(Y; f)$ in an obvious way.
We can also take restrictions of relations as well: for $Rsubseteq Y^n$ an $n$-ary relation on $Y$, its restriction to $X$ is $$R_X={(x_1,..., x_n)in X^n: (x_1,...,x_n)in R}.$$ Note that this can be more snappily represented as $R_X=Rcap X^n$. Similarly, conflating a function with its graph we have $f_X=fcap X^{n+1}$.
answered Nov 29 at 20:09
Noah Schweber
120k10146279
add a comment |
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If $Xsubseteq Y$ and $f:Y^nrightarrow Y$, we can look at how $f$ behaves on $X$: that is, look at the function $$f_X: X^nrightarrow Y.$$ Note that a priori this isn't a function from $X^n$ to $X$: $f$ might "go outside" of $X$, even if we just feed it things from $X$.
This $f_X$ is the restriction of $f$ to $X$ (I've never heard it called the "trace" before, but I wouldn't be too surprised if someone calls it that). If $ran(f_X)subseteq X$, then $(X;f_X$) is indeed a structure, and "lives inside" $(Y; f)$ in an obvious way.
We can also take restrictions of relations as well: for $Rsubseteq Y^n$ an $n$-ary relation on $Y$, its restriction to $X$ is $$R_X={(x_1,..., x_n)in X^n: (x_1,...,x_n)in R}.$$ Note that this can be more snappily represented as $R_X=Rcap X^n$. Similarly, conflating a function with its graph we have $f_X=fcap X^{n+1}$.
answered Nov 29 at 20:09
Noah Schweber
120k10146279
add a comment |
If $Xsubseteq Y$ and $f:Y^nrightarrow Y$, we can look at how $f$ behaves on $X$: that is, look at the function $$f_X: X^nrightarrow Y.$$ Note that a priori this isn't a function from $X^n$ to $X$: $f$ might "go outside" of $X$, even if we just feed it things from $X$.
This $f_X$ is the restriction of $f$ to $X$ (I've never heard it called the "trace" before, but I wouldn't be too surprised if someone calls it that). If $ran(f_X)subseteq X$, then $(X;f_X$) is indeed a structure, and "lives inside" $(Y; f)$ in an obvious way.
We can also take restrictions of relations as well: for $Rsubseteq Y^n$ an $n$-ary relation on $Y$, its restriction to $X$ is $$R_X={(x_1,..., x_n)in X^n: (x_1,...,x_n)in R}.$$ Note that this can be more snappily represented as $R_X=Rcap X^n$. Similarly, conflating a function with its graph we have $f_X=fcap X^{n+1}$.
answered Nov 29 at 20:09
Noah Schweber
120k10146279
add a comment |
If $Xsubseteq Y$ and $f:Y^nrightarrow Y$, we can look at how $f$ behaves on $X$: that is, look at the function $$f_X: X^nrightarrow Y.$$ Note that a priori this isn't a function from $X^n$ to $X$: $f$ might "go outside" of $X$, even if we just feed it things from $X$.
This $f_X$ is the restriction of $f$ to $X$ (I've never heard it called the "trace" before, but I wouldn't be too surprised if someone calls it that). If $ran(f_X)subseteq X$, then $(X;f_X$) is indeed a structure, and "lives inside" $(Y; f)$ in an obvious way.
We can also take restrictions of relations as well: for $Rsubseteq Y^n$ an $n$-ary relation on $Y$, its restriction to $X$ is $$R_X={(x_1,..., x_n)in X^n: (x_1,...,x_n)in R}.$$ Note that this can be more snappily represented as $R_X=Rcap X^n$. Similarly, conflating a function with its graph we have $f_X=fcap X^{n+1}$.
answered Nov 29 at 20:09
Noah Schweber
120k10146279
If $Xsubseteq Y$ and $f:Y^nrightarrow Y$, we can look at how $f$ behaves on $X$: that is, look at the function $$f_X: X^nrightarrow Y.$$ Note that a priori this isn't a function from $X^n$ to $X$: $f$ might "go outside" of $X$, even if we just feed it things from $X$.
This $f_X$ is the restriction of $f$ to $X$ (I've never heard it called the "trace" before, but I wouldn't be too surprised if someone calls it that). If $ran(f_X)subseteq X$, then $(X;f_X$) is indeed a structure, and "lives inside" $(Y; f)$ in an obvious way.
We can also take restrictions of relations as well: for $Rsubseteq Y^n$ an $n$-ary relation on $Y$, its restriction to $X$ is $$R_X={(x_1,..., x_n)in X^n: (x_1,...,x_n)in R}.$$ Note that this can be more snappily represented as $R_X=Rcap X^n$. Similarly, conflating a function with its graph we have $f_X=fcap X^{n+1}$.
answered Nov 29 at 20:09
Noah Schweber
120k10146279
answered Nov 29 at 20:09
Noah Schweber
120k10146279
answered Nov 29 at 20:09
Noah Schweber
120k10146279
answered Nov 29 at 20:09
Noah Schweber
120k10146279
120k10146279
add a comment |
add a comment |
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6
Restrictions. $ $
– Asaf Karagila♦
Nov 29 at 0:10