Abstract algebra / binary operation

The problem statement says that sigma(a)=ua. That's all it says. You can't assume that u is the inverse of a (it's not necessarily true) or that sigma is the identity (it's not necessarily true either).
  • #1
hsong9
80
1

Homework Statement


a. In each case a binary operation * is given on a set M. Decide whether it is commutative or associative, whether an identity exists, and find the units.
M=N(natrual); m*n = max(m,n)

b. If M is a moniod and u in M, let sigma: M -> M be defined by sigma(a) = ua for all a in M.
(a) show that sigma is a bijection if and only if u is a unit.
(b) If u is a unit, describe the inverse mapping sigma^-1: M -> M


Homework Equations


The Attempt at a Solution


In a, I know it is commutative and associative. I'm not sure identity and unit.
max(m,0) = always m, so 0 is identity, right?? how about unit? 0 is also unit?

in b, if u is a unit, sigma (a) = ua is gonne be identity or a?
Actually, I'm confusing about the concept of unit.

Thanks.
 
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  • #2
hsong9 said:

The Attempt at a Solution


max(m,0) = always m, so 0 is identity, right??

Yes.

how about unit? 0 is also unit?

Ask yourself, "Does 0 have an inverse in this set?" That is, "Does there exist an element x in this set such that x*0=0?" I think the answer is pretty clear.

in b, if u is a unit, sigma (a) = ua is gonne be identity or a?

Why does it have to be either? A unit is just an invertible element in a set. For instance consider the real numbers under multiplication. All nonzero elements are units. So let a=5 and u=2. Then [itex]\sigma (5)=(2)(5)=10[/itex] which is neither the identity nor a.

Actually, I'm confusing about the concept of unit.

That's what definitions are for. Have you read the definition of "unit"?
 
  • #3
so..
in b, if u is a unit, sigma(a) = ua can be any number?
or u is inverse of a?
if u is inverse of a, sigma(a) = ua = 1..
this is not bijection.. right?
 
  • #4
hsong9 said:
so..
in b, if u is a unit, sigma(a) = ua can be any number?

Well, [itex]a[/itex] can be any element of the monoid. The proposition that [itex]ua[/itex] can be any element is one of the things you're supposed to prove.

or u is inverse of a?

Not necessarily. In fact, since [itex]a[/itex] need not be a unit there's no reason to think that [itex]a[/itex] even has an inverse.

You seem to be assuming several things that are not implied by the problem statement. You really have to focus only on what is written there, and don't add anything to it.
 

Related to Abstract algebra / binary operation

What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It focuses on the properties and relationships between these structures, rather than specific numerical calculations.

What is a binary operation?

A binary operation is a mathematical operation that takes two elements from a set and combines them to produce a single element in the same set. Examples of binary operations include addition, subtraction, multiplication, and division.

What are some common examples of abstract algebraic structures?

Some common examples of abstract algebraic structures include groups, rings, fields, vector spaces, and algebras. These structures can be found in various areas of mathematics, such as number theory, geometry, and algebraic topology.

What is the significance of studying abstract algebra?

Studying abstract algebra helps us understand the fundamental principles and structures that underlie many mathematical concepts and theories. It also has many practical applications in fields such as physics, computer science, and cryptography.

What are some real-world applications of binary operations in abstract algebra?

Binary operations have many real-world applications, such as in computer science for coding and decoding data, in physics for calculating forces and motion, and in economics for analyzing financial transactions. They also play a crucial role in cryptography for creating secure communication channels.

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