Have a magma structure when “if the set of integers with respect to subtraction is not a group”? [closed]
$begingroup$
I have a 3 answers but nobody return me a mathematical structure/category name when I try to classify "the set of integers with respect to subtraction is not a group"
1) Subtraction of integers (and subtraction in general) is just addition with an inverse.
2) In almost every group the operation $gh^{-1}$ is not a group operation since it is not associative. The exception are groups where $h^{-1} = h$. So there is no needed extra structure; the inverse works just fine as it is
3rd answer is more complicated
3) “A group where I’m telling you the group’s subtraction operation rather than the group’s addition operation”.
That is, among many other ways to axiomatize it, one way to define a group is as any two abstract operations $+$ and $−$ on an inhabited set such that the following equations hold in general:
$a+(b+c)=(a+b)+c$
$a+(b−c)=(a+b)−c$
$a−(b+c)=(a−c)−b$ [Note the order reversal here!]
$a−(b−c)=(a+c)−b$ [And again here!]
$(a+b)−b=a$
$(a−b)+b=a$
$(a−a)+b=b$
It turns out that if two groups have the same $+$, they automatically agree on everything else as well. And people often like to refer to groups just by referring to their + operator; so much so that people will call this “the group operation” or such things.
But it also turns out that if two groups have the same $−$, they automatically agree on everything else as well. The − formalized above is an operation just as well available in any group. And so you could just as well refer to groups by referring to their $−$ operator. This is a different way of specifying a group than by specifying its $+$ operator, but it works just as well.
$+$ and $−$ don’t have the same properties as each other, but they’re both equally a part of any group, and they both equally determine everything there is to say about a group.
So the integers with respect to subtraction can be thought of as an instance of this $−$ structure. Which amounts to the same thing as a group, but is described with respect to a different operation. If you like, call it a “soup”, and note that every group comes with a particular corresponding soup and vice versa.
Incidentally, I’ve described everything here using the language of + and −, but a very strong convention is to only use these names when dealing with groups which are commutative (that is, where a+b=b+a), and to otherwise speak in the language of multiplication and division instead. Still, to avoid confusion in this context, it felt best to simply use additive language.
group-theory magma
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
$endgroup$
closed as unclear what you're asking by Shaun, Slade, Derek Holt, Paul Frost, Namaste Dec 23 '18 at 16:25
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I have a 3 answers but nobody return me a mathematical structure/category name when I try to classify "the set of integers with respect to subtraction is not a group"
1) Subtraction of integers (and subtraction in general) is just addition with an inverse.
2) In almost every group the operation $gh^{-1}$ is not a group operation since it is not associative. The exception are groups where $h^{-1} = h$. So there is no needed extra structure; the inverse works just fine as it is
3rd answer is more complicated
3) “A group where I’m telling you the group’s subtraction operation rather than the group’s addition operation”.
That is, among many other ways to axiomatize it, one way to define a group is as any two abstract operations $+$ and $−$ on an inhabited set such that the following equations hold in general:
$a+(b+c)=(a+b)+c$
$a+(b−c)=(a+b)−c$
$a−(b+c)=(a−c)−b$ [Note the order reversal here!]
$a−(b−c)=(a+c)−b$ [And again here!]
$(a+b)−b=a$
$(a−b)+b=a$
$(a−a)+b=b$
It turns out that if two groups have the same $+$, they automatically agree on everything else as well. And people often like to refer to groups just by referring to their + operator; so much so that people will call this “the group operation” or such things.
But it also turns out that if two groups have the same $−$, they automatically agree on everything else as well. The − formalized above is an operation just as well available in any group. And so you could just as well refer to groups by referring to their $−$ operator. This is a different way of specifying a group than by specifying its $+$ operator, but it works just as well.
$+$ and $−$ don’t have the same properties as each other, but they’re both equally a part of any group, and they both equally determine everything there is to say about a group.
So the integers with respect to subtraction can be thought of as an instance of this $−$ structure. Which amounts to the same thing as a group, but is described with respect to a different operation. If you like, call it a “soup”, and note that every group comes with a particular corresponding soup and vice versa.
Incidentally, I’ve described everything here using the language of + and −, but a very strong convention is to only use these names when dealing with groups which are commutative (that is, where a+b=b+a), and to otherwise speak in the language of multiplication and division instead. Still, to avoid confusion in this context, it felt best to simply use additive language.
group-theory magma
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
$endgroup$
closed as unclear what you're asking by Shaun, Slade, Derek Holt, Paul Frost, Namaste Dec 23 '18 at 16:25
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
What's the question here?
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 13:34
1
$begingroup$
What exactly is your question?
$endgroup$
– Rob Arthan
Dec 23 '18 at 13:34
$begingroup$
The question is "If the set of integers, with respect to subtraction, is not a group, what type of mathematical structure do we need to talk about?" A magma, a Kripke model ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:47
add a comment |
$begingroup$
I have a 3 answers but nobody return me a mathematical structure/category name when I try to classify "the set of integers with respect to subtraction is not a group"
1) Subtraction of integers (and subtraction in general) is just addition with an inverse.
2) In almost every group the operation $gh^{-1}$ is not a group operation since it is not associative. The exception are groups where $h^{-1} = h$. So there is no needed extra structure; the inverse works just fine as it is
3rd answer is more complicated
3) “A group where I’m telling you the group’s subtraction operation rather than the group’s addition operation”.
That is, among many other ways to axiomatize it, one way to define a group is as any two abstract operations $+$ and $−$ on an inhabited set such that the following equations hold in general:
$a+(b+c)=(a+b)+c$
$a+(b−c)=(a+b)−c$
$a−(b+c)=(a−c)−b$ [Note the order reversal here!]
$a−(b−c)=(a+c)−b$ [And again here!]
$(a+b)−b=a$
$(a−b)+b=a$
$(a−a)+b=b$
It turns out that if two groups have the same $+$, they automatically agree on everything else as well. And people often like to refer to groups just by referring to their + operator; so much so that people will call this “the group operation” or such things.
But it also turns out that if two groups have the same $−$, they automatically agree on everything else as well. The − formalized above is an operation just as well available in any group. And so you could just as well refer to groups by referring to their $−$ operator. This is a different way of specifying a group than by specifying its $+$ operator, but it works just as well.
$+$ and $−$ don’t have the same properties as each other, but they’re both equally a part of any group, and they both equally determine everything there is to say about a group.
So the integers with respect to subtraction can be thought of as an instance of this $−$ structure. Which amounts to the same thing as a group, but is described with respect to a different operation. If you like, call it a “soup”, and note that every group comes with a particular corresponding soup and vice versa.
Incidentally, I’ve described everything here using the language of + and −, but a very strong convention is to only use these names when dealing with groups which are commutative (that is, where a+b=b+a), and to otherwise speak in the language of multiplication and division instead. Still, to avoid confusion in this context, it felt best to simply use additive language.
group-theory magma
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
$endgroup$
I have a 3 answers but nobody return me a mathematical structure/category name when I try to classify "the set of integers with respect to subtraction is not a group"
1) Subtraction of integers (and subtraction in general) is just addition with an inverse.
2) In almost every group the operation $gh^{-1}$ is not a group operation since it is not associative. The exception are groups where $h^{-1} = h$. So there is no needed extra structure; the inverse works just fine as it is
3rd answer is more complicated
3) “A group where I’m telling you the group’s subtraction operation rather than the group’s addition operation”.
That is, among many other ways to axiomatize it, one way to define a group is as any two abstract operations $+$ and $−$ on an inhabited set such that the following equations hold in general:
$a+(b+c)=(a+b)+c$
$a+(b−c)=(a+b)−c$
$a−(b+c)=(a−c)−b$ [Note the order reversal here!]
$a−(b−c)=(a+c)−b$ [And again here!]
$(a+b)−b=a$
$(a−b)+b=a$
$(a−a)+b=b$
It turns out that if two groups have the same $+$, they automatically agree on everything else as well. And people often like to refer to groups just by referring to their + operator; so much so that people will call this “the group operation” or such things.
But it also turns out that if two groups have the same $−$, they automatically agree on everything else as well. The − formalized above is an operation just as well available in any group. And so you could just as well refer to groups by referring to their $−$ operator. This is a different way of specifying a group than by specifying its $+$ operator, but it works just as well.
$+$ and $−$ don’t have the same properties as each other, but they’re both equally a part of any group, and they both equally determine everything there is to say about a group.
So the integers with respect to subtraction can be thought of as an instance of this $−$ structure. Which amounts to the same thing as a group, but is described with respect to a different operation. If you like, call it a “soup”, and note that every group comes with a particular corresponding soup and vice versa.
Incidentally, I’ve described everything here using the language of + and −, but a very strong convention is to only use these names when dealing with groups which are commutative (that is, where a+b=b+a), and to otherwise speak in the language of multiplication and division instead. Still, to avoid confusion in this context, it felt best to simply use additive language.
group-theory magma
group-theory magma
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
asked Dec 23 '18 at 13:22
Peter LongPeter Long
194
194
closed as unclear what you're asking by Shaun, Slade, Derek Holt, Paul Frost, Namaste Dec 23 '18 at 16:25
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Shaun, Slade, Derek Holt, Paul Frost, Namaste Dec 23 '18 at 16:25
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
What's the question here?
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 13:34
1
$begingroup$
What exactly is your question?
$endgroup$
– Rob Arthan
Dec 23 '18 at 13:34
$begingroup$
The question is "If the set of integers, with respect to subtraction, is not a group, what type of mathematical structure do we need to talk about?" A magma, a Kripke model ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:47
add a comment |
2
$begingroup$
What's the question here?
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 13:34
1
$begingroup$
What exactly is your question?
$endgroup$
– Rob Arthan
Dec 23 '18 at 13:34
$begingroup$
The question is "If the set of integers, with respect to subtraction, is not a group, what type of mathematical structure do we need to talk about?" A magma, a Kripke model ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:47
2
2
$begingroup$
What's the question here?
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 13:34
$begingroup$
What's the question here?
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 13:34
1
1
$begingroup$
What exactly is your question?
$endgroup$
– Rob Arthan
Dec 23 '18 at 13:34
$begingroup$
What exactly is your question?
$endgroup$
– Rob Arthan
Dec 23 '18 at 13:34
$begingroup$
The question is "If the set of integers, with respect to subtraction, is not a group, what type of mathematical structure do we need to talk about?" A magma, a Kripke model ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:47
$begingroup$
The question is "If the set of integers, with respect to subtraction, is not a group, what type of mathematical structure do we need to talk about?" A magma, a Kripke model ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:47
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
I suspect that the question being asked here is if there exists a theory about sets with one binary operation that is closed and has inverses; if that is the case then the answer is yes, and they are called quasigroups.
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
$endgroup$
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
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– quid♦
Dec 23 '18 at 17:18
add a comment |
$begingroup$
The set of integers $mathbb{Z}$ with operation $-$ is a magma, a set with a binary operation.
As $-$ is not associative $(1-2)-3 = -4 neq 2= 1-(2-3)$, we don't have a semigroup or a monoid, even though it has an identity $0$, and even inverses.
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
$endgroup$
$begingroup$
But also a set of integers Z with operation $+$ is a magma? How can I tell if a function on a generic magma has $+$ or $-$ operation ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:51
$begingroup$
@PeterLong you cannot. The operation is just a function. But $+$ is associative and $-$ is not. With $+$ we even have a group.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 13:53
$begingroup$
mm..ok, but if − is on an inhabited set does the kripke model have anything to do with it? Because the 'position' in which magma is it seems to me the condition "X is nonempty" without satisfying "X is inhabited" (Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "X is nonempty" implies "X is inhabited")
$endgroup$
– Peter Long
Dec 23 '18 at 13:58
$begingroup$
@PeterLong I don’t see what Kripke models or intuitionistic logic has to do with such a basic abstract algebra question.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 14:58
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I suspect that the question being asked here is if there exists a theory about sets with one binary operation that is closed and has inverses; if that is the case then the answer is yes, and they are called quasigroups.
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
$endgroup$
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– quid♦
Dec 23 '18 at 17:18
add a comment |
$begingroup$
I suspect that the question being asked here is if there exists a theory about sets with one binary operation that is closed and has inverses; if that is the case then the answer is yes, and they are called quasigroups.
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
$endgroup$
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– quid♦
Dec 23 '18 at 17:18
add a comment |
$begingroup$
I suspect that the question being asked here is if there exists a theory about sets with one binary operation that is closed and has inverses; if that is the case then the answer is yes, and they are called quasigroups.
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
$endgroup$
I suspect that the question being asked here is if there exists a theory about sets with one binary operation that is closed and has inverses; if that is the case then the answer is yes, and they are called quasigroups.
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
answered Dec 23 '18 at 13:47
RandomNumberGuyRandomNumberGuy
662
662
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– quid♦
Dec 23 '18 at 17:18
add a comment |
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– quid♦
Dec 23 '18 at 17:18
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– quid♦
Dec 23 '18 at 17:18
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– quid♦
Dec 23 '18 at 17:18
add a comment |
$begingroup$
The set of integers $mathbb{Z}$ with operation $-$ is a magma, a set with a binary operation.
As $-$ is not associative $(1-2)-3 = -4 neq 2= 1-(2-3)$, we don't have a semigroup or a monoid, even though it has an identity $0$, and even inverses.
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
$endgroup$
$begingroup$
But also a set of integers Z with operation $+$ is a magma? How can I tell if a function on a generic magma has $+$ or $-$ operation ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:51
$begingroup$
@PeterLong you cannot. The operation is just a function. But $+$ is associative and $-$ is not. With $+$ we even have a group.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 13:53
$begingroup$
mm..ok, but if − is on an inhabited set does the kripke model have anything to do with it? Because the 'position' in which magma is it seems to me the condition "X is nonempty" without satisfying "X is inhabited" (Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "X is nonempty" implies "X is inhabited")
$endgroup$
– Peter Long
Dec 23 '18 at 13:58
$begingroup$
@PeterLong I don’t see what Kripke models or intuitionistic logic has to do with such a basic abstract algebra question.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 14:58
add a comment |
$begingroup$
The set of integers $mathbb{Z}$ with operation $-$ is a magma, a set with a binary operation.
As $-$ is not associative $(1-2)-3 = -4 neq 2= 1-(2-3)$, we don't have a semigroup or a monoid, even though it has an identity $0$, and even inverses.
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
$endgroup$
$begingroup$
But also a set of integers Z with operation $+$ is a magma? How can I tell if a function on a generic magma has $+$ or $-$ operation ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:51
$begingroup$
@PeterLong you cannot. The operation is just a function. But $+$ is associative and $-$ is not. With $+$ we even have a group.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 13:53
$begingroup$
mm..ok, but if − is on an inhabited set does the kripke model have anything to do with it? Because the 'position' in which magma is it seems to me the condition "X is nonempty" without satisfying "X is inhabited" (Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "X is nonempty" implies "X is inhabited")
$endgroup$
– Peter Long
Dec 23 '18 at 13:58
$begingroup$
@PeterLong I don’t see what Kripke models or intuitionistic logic has to do with such a basic abstract algebra question.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 14:58
add a comment |
$begingroup$
The set of integers $mathbb{Z}$ with operation $-$ is a magma, a set with a binary operation.
As $-$ is not associative $(1-2)-3 = -4 neq 2= 1-(2-3)$, we don't have a semigroup or a monoid, even though it has an identity $0$, and even inverses.
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
$endgroup$
The set of integers $mathbb{Z}$ with operation $-$ is a magma, a set with a binary operation.
As $-$ is not associative $(1-2)-3 = -4 neq 2= 1-(2-3)$, we don't have a semigroup or a monoid, even though it has an identity $0$, and even inverses.
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
answered Dec 23 '18 at 13:48
Henno BrandsmaHenno Brandsma
117k350128
117k350128
$begingroup$
But also a set of integers Z with operation $+$ is a magma? How can I tell if a function on a generic magma has $+$ or $-$ operation ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:51
$begingroup$
@PeterLong you cannot. The operation is just a function. But $+$ is associative and $-$ is not. With $+$ we even have a group.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 13:53
$begingroup$
mm..ok, but if − is on an inhabited set does the kripke model have anything to do with it? Because the 'position' in which magma is it seems to me the condition "X is nonempty" without satisfying "X is inhabited" (Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "X is nonempty" implies "X is inhabited")
$endgroup$
– Peter Long
Dec 23 '18 at 13:58
$begingroup$
@PeterLong I don’t see what Kripke models or intuitionistic logic has to do with such a basic abstract algebra question.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 14:58
add a comment |
$begingroup$
But also a set of integers Z with operation $+$ is a magma? How can I tell if a function on a generic magma has $+$ or $-$ operation ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:51
$begingroup$
@PeterLong you cannot. The operation is just a function. But $+$ is associative and $-$ is not. With $+$ we even have a group.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 13:53
$begingroup$
mm..ok, but if − is on an inhabited set does the kripke model have anything to do with it? Because the 'position' in which magma is it seems to me the condition "X is nonempty" without satisfying "X is inhabited" (Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "X is nonempty" implies "X is inhabited")
$endgroup$
– Peter Long
Dec 23 '18 at 13:58
$begingroup$
@PeterLong I don’t see what Kripke models or intuitionistic logic has to do with such a basic abstract algebra question.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 14:58
$begingroup$
But also a set of integers Z with operation $+$ is a magma? How can I tell if a function on a generic magma has $+$ or $-$ operation ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:51
$begingroup$
But also a set of integers Z with operation $+$ is a magma? How can I tell if a function on a generic magma has $+$ or $-$ operation ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:51
$begingroup$
@PeterLong you cannot. The operation is just a function. But $+$ is associative and $-$ is not. With $+$ we even have a group.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 13:53
$begingroup$
@PeterLong you cannot. The operation is just a function. But $+$ is associative and $-$ is not. With $+$ we even have a group.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 13:53
$begingroup$
mm..ok, but if − is on an inhabited set does the kripke model have anything to do with it? Because the 'position' in which magma is it seems to me the condition "X is nonempty" without satisfying "X is inhabited" (Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "X is nonempty" implies "X is inhabited")
$endgroup$
– Peter Long
Dec 23 '18 at 13:58
$begingroup$
mm..ok, but if − is on an inhabited set does the kripke model have anything to do with it? Because the 'position' in which magma is it seems to me the condition "X is nonempty" without satisfying "X is inhabited" (Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "X is nonempty" implies "X is inhabited")
$endgroup$
– Peter Long
Dec 23 '18 at 13:58
$begingroup$
@PeterLong I don’t see what Kripke models or intuitionistic logic has to do with such a basic abstract algebra question.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 14:58
$begingroup$
@PeterLong I don’t see what Kripke models or intuitionistic logic has to do with such a basic abstract algebra question.
$endgroup$
– Henno Brandsma
Dec 23 '18 at 14:58
add a comment |
2
$begingroup$
What's the question here?
$endgroup$
– Lord Shark the Unknown
Dec 23 '18 at 13:34
1
$begingroup$
What exactly is your question?
$endgroup$
– Rob Arthan
Dec 23 '18 at 13:34
$begingroup$
The question is "If the set of integers, with respect to subtraction, is not a group, what type of mathematical structure do we need to talk about?" A magma, a Kripke model ?
$endgroup$
– Peter Long
Dec 23 '18 at 13:47