What does structure mean in maths
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Steven G
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There are many structures in mathematics.
One structure is called a group.
A group has a set, G, and an operation on this set called *.
A group has the following properties
1) If a and b are in G, then a*b is in G
2) If a, b and c are in G, then (a*b)*c = a*(b*c)
3) There is an element e in G so that a*e = e*a = a for all a in G
4) For every a in G there exist a' in G so that a*a' = a'*a=e
Does this look familiar to anything that you already learned (It should!)?
One structure is called a group.
A group has a set, G, and an operation on this set called *.
A group has the following properties
1) If a and b are in G, then a*b is in G
2) If a, b and c are in G, then (a*b)*c = a*(b*c)
3) There is an element e in G so that a*e = e*a = a for all a in G
4) For every a in G there exist a' in G so that a*a' = a'*a=e
Does this look familiar to anything that you already learned (It should!)?
topsquark
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I'll expand a bit on Steven G's excellent example. We common take "simple" concepts such as the integers and add things to them, constructing different "objects" such as the rational numbers by defining division. You could call this process "building a structure."
-Dan
Dr.Peterson
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It might mean several different things. Can you show us the context in which you saw the word?
In the context of abstract algebra, a “structure” means a set and one or more operations defined on members of the set.
Between college and graduate school, I took a course in abstract algebra. It was fascinating, but I found it very difficult. The primary reason for my difficulty was that I was unfamiliar with almost all the examples so they did not help me see what was being generalized. I’d suggest learning at least matrix algebra before taking on abstract algebra.
To see how the word “structure” is used in abstract algebra, see
and
What is difference between a ring and a field?
The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra
math.stackexchange.com
So semi-groups, monoids, groups, rings, and fields are all types of structure. Notice that, for example, there are different varieties of rings.
The practical importance of abstract algebra is that a theorem about some type of ring applies to anything that meets the definition of that type of ring. So, if a physicist discovers something that can be described as an operation and a set of something and that description fits the definition, the physicist has a whole algebra that can be used to analyze this discovery.
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Data structure in computer science refers to the "way information (data)" are stored - so that those can be retrieved quickly by a program.
Steven G
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Does this look familiar to anything that you already learned (It should!)? Like in k-5th grade?
blamocur
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Pretty short and informative: https://en.wikipedia.org/wiki/Mathematical_structure
Steven G
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To OP.
My structure above (of a group) should look familiar.
If we consider the set of Integers and addition as the operation, then we have a group.
If you add to integers you get an integer.
Integers have the associative law under addition.
0 is an integer and 0+a=a+0 for any a in the integers.
If a any integer, then there is an integer -a such that a+(-a) = 0.
The real numbers without 0 with multiplication is also a group. Why can't we have 0?
My structure above (of a group) should look familiar.
If we consider the set of Integers and addition as the operation, then we have a group.
If you add to integers you get an integer.
Integers have the associative law under addition.
0 is an integer and 0+a=a+0 for any a in the integers.
If a any integer, then there is an integer -a such that a+(-a) = 0.
The real numbers without 0 with multiplication is also a group. Why can't we have 0?
@Steven G
The rational numbers under the operations of addition and multiplication form an ordered field. In fact, it is obvious that the various algebraic structures that this thread has mentioned were built up by analogy to the rational and real numbers, the four basic arithmetic operations, and the order relations exemplified by the rational and real numbers.
The intellectual interest of abstract algebra is to see that algebraic structures differing from those we have been familiar with since grade school are logically possible, that any system that can be made to fit with a structure previously studied by abstract algebra already has an appropriate algebra, and that arithmetic and elementary algebra are monumentally useful because so much of physical reality can be mapped on to the structure of an ordered field.
But it seems to me that the pedagogic issue is to show examples that differ from what seems obvious from arithmetic and that have obvious practical application. Solving Diophantine equations might be such an example. The positive integers under addition form a commutative semi-group and under multiplication form a commutative monoid. This is a much sparser structure than the field that is studied in arithmetic and elementary algebra, which explains why solving Diophantine problems is hard.
Based on my own experience, a major difficulty in learning abstract algebra is a lack of examples that obviously arise in the physical world.
"When we think of a structure in the everyday sense, we might think of buildings, houses, and bridges. We may also think of a structure as a more abstract object involving some form of complex organization."
truebeautyofmath.com
What does "complex organisation" mean here?
Thanks.
What Is A Mathematical Structure?
In the post “What is math?”, we described mathematics as the art of creating and exploring mathematical structures. It is not unlikely, however, that the reader is slightly unfamiliar …
truebeautyofmath.com
Thanks.
As Dr. Peterson said earlier, the word “structure” may have different meanings in different branches of mathematics. In algebra, it means a set, one or more operations on the elements of a set, some restrictions on those operations, and perhaps an order relation among the elements of the set.
So a magma is a structure that has a non-empty set S, a binary operation denoted by @ that can be applied to any two elements of S with the following limitation:
[math]a,\ b \in \mathbb S \implies a \ @ \ b \in \mathbb S.[/math]
Notice that the odd integers with the arithmetic operation of addition do not form a magma.
A more complex structure is a semi-group, which is a magma with an additional limitation, namely
[math]a, \ b, \ c \in \mathbb S \implies (a \ @ \ b) \ @ \ c = a \ @ \ (b \ @ \ c).[/math]
An even more complex structure is a monoid, which is a semi-group with an additional limitation, namely
[math]\exists \ i_@ \in \mathbb S \text { such that } a \ @ \ i_@ = a \text { for any } a \in \mathbb S.[/math]
Notice that the even integers with the arithmetic operation of multiplication form a semi-group, but not a monoid.
We can build a more complex structure called a group by adding one more restriction to a monoid, namely
[math]a \in \mathbb S \implies \exists \ b \in \mathbb S \text { such that } a \ @ \ b = i_@.[/math]
And we can put together a still yet more complex structure called an abelian group by adding still yet another restriction, namely
[math]a, \ b \in \mathbb S \implies a \ @ \ b = b \ @ \ a.[/math]
And we can go on, especially by having more than one operation.
So maybe now you can see what is meant by a structure in algebra and how more complex structures are built from simpler structures.
So a magma is a structure that has a non-empty set S, a binary operation denoted by @ that can be applied to any two elements of S with the following limitation:
[math]a,\ b \in \mathbb S \implies a \ @ \ b \in \mathbb S.[/math]
Notice that the odd integers with the arithmetic operation of addition do not form a magma.
A more complex structure is a semi-group, which is a magma with an additional limitation, namely
[math]a, \ b, \ c \in \mathbb S \implies (a \ @ \ b) \ @ \ c = a \ @ \ (b \ @ \ c).[/math]
An even more complex structure is a monoid, which is a semi-group with an additional limitation, namely
[math]\exists \ i_@ \in \mathbb S \text { such that } a \ @ \ i_@ = a \text { for any } a \in \mathbb S.[/math]
Notice that the even integers with the arithmetic operation of multiplication form a semi-group, but not a monoid.
We can build a more complex structure called a group by adding one more restriction to a monoid, namely
[math]a \in \mathbb S \implies \exists \ b \in \mathbb S \text { such that } a \ @ \ b = i_@.[/math]
And we can put together a still yet more complex structure called an abelian group by adding still yet another restriction, namely
[math]a, \ b \in \mathbb S \implies a \ @ \ b = b \ @ \ a.[/math]
And we can go on, especially by having more than one operation.
So maybe now you can see what is meant by a structure in algebra and how more complex structures are built from simpler structures.