Tour:Invertible implies cancellative in monoid
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This article adapts material from the main article: invertible implies cancellative in monoid
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WHAT YOU NEED TO DO:
- Understand the statement below; try proving it yourself
- Understand the proof, and the crucial way in which it relies on associativity
PONDER:
- What happens if we remove the assumption of associativity? Can you cook up a magma (set with binary operation) having a neutral element, where an element has a left inverse but is not cancellative?
Statement
In a monoid (a set with an associative binary operation possessing a multiplicative identity element) the following are true:
- Any left invertible element (element having a left inverse) is left cancellative.
- Any right invertible element (element having a right inverse) is right cancellative.
- Any invertible element is cancellative.
Proof
We'll give here the proof for left invertible and left cancellative. An analogous proof works for right invertible and right cancellative.
Given: A monoid with binary operation
, and identity element (also called neutral element)
.
has a left inverse
(i.e. an element
)
To prove: is left-cancellative: whenever
are such that
, then
.
Proof: We start with:
Left-multiply both sides by :
Use associativity:
We now use that is the identity element, to conclude that
.
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