Is a Cancellative Semigroup the Same as a Group?

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Discussion Overview

The discussion revolves around the properties of cancellative semigroups and their relationship to groups. Participants explore whether a set that is closed, associative, and satisfies the cancellation laws can be classified as a group.

Discussion Character

  • Debate/contested, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asserts that if a set is closed and associative with respect to an operation and both cancellation laws hold, it should be proven that the set is a group with respect to that operation.
  • Another participant questions whether this is a homework problem and asks for an attempt at a solution to be shown.
  • A participant expresses gratitude, indicating that they have completed the problem.
  • One participant challenges the initial assertion, stating that the set of positive integers under multiplication is closed, associative, and satisfies both cancellation laws, yet does not form a group.
  • Another participant adds that the integers greater than or equal to 2 under multiplication do not even form a monoid, introducing the term "cancellative semigroup" to describe such structures.

Areas of Agreement / Disagreement

Participants express disagreement regarding the initial claim that a cancellative semigroup must be a group. Multiple competing views remain, particularly concerning the examples provided and their implications.

Contextual Notes

There are limitations in the discussion regarding the definitions of groups and cancellative semigroups, as well as the implications of the examples cited. The discussion does not resolve these complexities.

prashantgolu
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if a set is closed and associative with respect to an operation * and both cancllation laws hold...prove that the set is a group wrt *.
 
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Is this homework? Have you made an attempt at the problem? Show us what you have.
 
thnx..this one is done...
 
It looks to me like you are trying to prove something that is NOT TRUE. For example, the set of positive integers is closed under ordinary mulitplication which is associative and both cancellation laws hold. But this is not a group.
 
Ah, indeed. And if you take the integers ≥ 2 under multiplication, then you don't even get a monoid. Apparently such a thing is called a cancellative semigroup.
 

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