Today I want to talk about what an algebraic structure is.
And why I want to talk about this strange thing ?, simple, because it is good to know the bases. Because when we do not know what to do or how to develop something, we must always go to definition and apply it. Because algebra is necessary although very hated, simply by ignorance.
As I said, a structure is something that defines a reality and that, therefore, are the basis of something.
An algebraic structure, is the basis of an algebraic system. It consists of a series of elements (they can be numbers, they can be types of chestnuts, they can be people) different from each other and what we consider an operation between that group of elements.
An operation between the group of elements is a function that allows us to move from one to another element, either for example the sum (which is the addiction of one element and another) or the change of jersey between people.
The operation, to form an algebraic structure, must be at least one, just as the elements must be at least one as well, but that does not mean that they can be more or even infinite.
Different algebraic structures exist according to the operations and the result that we obtain. For example, if the sum of two elements gives you a third of the same group or that the change of jersey between two people gives us new elements of the same group. In fact, the operations can give us elements of another group different from the one that we have as base, being all this correct and valid.
Depending on the result of the operations there are some laws classifications of algebraic structures and therefore a way to classify them.
For example, those that have the operation that is, the negative of the operation that is, that the order of the operation does not alter the result and that any of the above of another element of the same set or group is called a commutative group.
The example of the people and the change of the jersey. If two change the jersey, you can change the jersey back to the origin (inverse), it does not matter if three change the jersey because the final result (where the sweaters are) is the same, we would give the jersey change is A commutative group. It’s Fun!.
That is, we can consider a group of people and the jersey change operation as an algebraic structure. This helps us to observe that math is everywhere, even in a clothing store, and it does not have to do with numbers if it is imaginative.