Is a Monoid the Same as a Semi-group in Topology?

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In summary: In my experience, semi-group means associative binary operation, and monoid means associative binary operation with identity element.
  • #1
quasar987
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My topology teacher appears to call a monoid a set with an associative binary operation, but with no identity element. According to wiki, this is the definition of a semi-group, although they remark that some authors define semi-groups as having an identity (i.e. synonymously to monoid). But they don't say on the monoid article that some authors take monoid to mean an associative magma(groupoid) with no identity.

So, does my teacher simply has the definitions mixed up or do some authors effectively call 'monoid' an associative magma(groupoid)?

I wanted to ask here before throwing the "Sir professor, according to wikipedia, you're wrong" at him. I'm sure that's understandable. :rolleyes:
 
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  • #2
IMOH, arguments over terminology are politically dangerous, only attempt this if you are on good terms with your professor, or want to start a fight.

In my experience, semi-group means associative binary operation, and monoid means associative binary operation with identity element. Wolfram's mathworld agrees: http://mathworld.wolfram.com/Monoid.html, I usually trust Wolfram for ORTHODOX definitions, wikipedia is good at bringing in side issues and lesser known usage (ok, that's my subjective opinion).
 
  • #3
Yeah, I wouldn't recommend telling him he's "wrong", but you might want to tell him that you've seen it mean something else in (many) other standard references. Like, for example, in Rotman's Theory of Groups.
 
  • #4
You could tell him. Though he's likely to be unimpressed that you used Wiki as a source, and even less impressed that you're worrying about this than actually learning the course.

It's just a name, and mostly unimportant. It is the definition that is important. Ok, it might cause you some confusion when looking in other textbooks.
 
  • #5
ecurbian said:
IMOH, arguments over terminology are politically dangerous, only attempt this if you are on good terms with your professor, or want to start a fight.
Quite a lot of even modern mathematical definitions are not standardised with subtle and not so subtle differences cropping up all over the place. Many books will even use different "derivation trees" if you will to arrive at some concepts and objects earlier or later than others would.
 

1. What are semi-groups and monoids?

Semi-groups and monoids are algebraic structures that consist of a set of elements and an operation that combines any two elements to produce a third element. The main difference between the two is that a monoid has an identity element, while a semi-group does not necessarily have one.

2. What are some real-world applications of semi-groups and monoids?

Semi-groups and monoids are used in many areas of mathematics and computer science, such as in the study of abstract algebra, automata theory, and formal languages. They also have practical applications in fields such as coding theory, cryptography, and data compression.

3. How do semi-groups and monoids differ from groups?

All three structures share the same properties of closure, associativity, and identity, but groups also have the added property of invertibility. This means that for every element in a group, there exists an inverse element that, when combined, results in the identity element. Semi-groups and monoids do not have this property.

4. Can semi-groups and monoids have more than one operation?

Yes, semi-groups and monoids can have multiple operations as long as they still adhere to the defining properties. For example, a monoid may have both addition and multiplication as operations, as long as they are still associative and there is an identity element for each operation.

5. How are semi-groups and monoids related to other algebraic structures?

Semi-groups and monoids are considered "simpler" algebraic structures compared to groups and rings, as they have fewer properties that need to be satisfied. They can also be seen as building blocks for more complex structures, such as groups, rings, and fields, which often incorporate the properties of semi-groups and monoids as a foundation.

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