What are Quotient Spaces and How Are They Used in Algebra and Topology?

[flyerpower]

I'm having some troubles understanding the concepts of quotient algebra.
May someone explain me what exactly they are, giving some concrete examples?

I know that a quotient set is the set of all equivalence classes, but it sounds very vague for me and i can't make the analogy with quotient spaces, or quotient groups.

 

[Deveno]

what do you mean by "quotient algebra"? there are different kinds of algebras:

1. universal algebras (defined by a set and a signature of arity, and identities involving n-ary operations, etc.)
2. associative algebras (vector spaces with an associative ring structure).

 

[Fredrik]

what do you mean by "quotient algebra"?

If A is an algebra and I is an ideal in A, the quotient algebra A/I is defined in the following way. Define a relation ~ on A by saying that x~y if x-y is a member of I. It's easy to show that ~ is an equivalence relation. The equivalence class that contains x is denoted by x+I. The set of all equivalence classes is given the structure of an algebra by the definitions

(x+I)+(y+I)=(x+y)+I
a(x+I)=(ax)+I
(x+I)(y+I)=xy+I

 

[Deveno]

If A is an algebra and I is an ideal in A, the quotient algebra A/I is defined in the following way. Define a relation ~ on A by saying that x~y if x-y is a member of I. It's easy to show that ~ is an equivalence relation. The equivalence class that contains x is denoted by x+I. The set of all equivalence classes is given the structure of an algebra by the definitions

(x+I)+(y+I)=(x+y)+I
a(x+I)=(ax)+I
(x+I)(y+I)=xy+I

yes, that follows by definition 2. in the algebras of type 1, you generally don't have ideals, but rather congruences. i was asking "which" definition of algebra he meant.

 

[blue_raver22]

A quotient space is a mathematical concept that is often used in algebra and topology. It is a way to construct a new space from an existing one by "identifying" certain points or elements in the original space. This process of identification is done by defining an equivalence relation on the original space.

To give a concrete example, let's consider the real numbers. We can define an equivalence relation on the real numbers by saying that two real numbers are equivalent if their difference is an integer. This means that, for example, 1 and 2 are equivalent because their difference is 1, which is an integer. Similarly, 3.5 and 4.5 are equivalent because their difference is 1, which is an integer.

Now, we can construct a quotient space by taking the real numbers and "identifying" all the equivalent points. This means that we are essentially grouping together all the points that are equivalent to each other and treating them as a single point in the quotient space. In this example, our quotient space would consist of just two points, 0 and 1, since all the other points in the real numbers are equivalent to either 0 or 1.

This concept can also be extended to other mathematical structures, such as groups and vector spaces. In a quotient group, we identify certain elements of the original group and treat them as a single element in the quotient group. This allows us to simplify the structure of the group and make it easier to work with. Similarly, in a quotient vector space, we identify certain vectors and treat them as a single vector, reducing the dimension of the space.

I hope this explanation helps to clarify the concept of quotient spaces for you. Please let me know if you have any further questions or need more clarification.