Cancellative set in a semiring that is not multiplicatively closed

In summary, a semigroup is a pair (R,op) where R is a set and op is a binary operation that is closed and associative. A commutative semigroup is a semigroup where the binary operation is commutative. A monoid is a semigroup with an identity element that satisfies the monoid and commutative semigroup laws. A commutative monoid is a semigroup that satisfies all three laws. A semiring (rig) is a triple (R,+,.) where (R,+) is a commutative monoid and (R,.) is a semigroup with distributivity. A cancellable element in a semiring is one where r+a = r+b implies a =
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icantadd
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Definition: A semigroup is a pair (R,op) where R is a set an op is a binary operation that is closed and associative. A commutative semigroup is a semigroup where op satisfies for all a,b in R, op(a,b) = op(b,a). A monoid is a semigroup where with an identity,e, for op, satisfying for all r in R, op(r,e)=op(e,r)=r. A commutative monoid is a semigroup satisfying the monoid and commutative semigroup laws.

Definition:
A semiring (rig) is a triple (R,+,.) where (R,+) is a commutative monoid, and (R,.) is a semigroup, and where distributivity holds;i.e.
a(b+c) = ab+ac and (b+c)a = ba+ca.

Definition:
Let R=(R,+,.) be a semiring. A cancellable element r satisfies
r+a = r+b implies a = b.
The set of all cancellable elements in R is denoted Can(R).

It is an easy exercise to show that Can(R) is a submonoid of (R,+). However, it does not seem to be multiplicatively closed. If anyone knows or can sketch a proof that would be great! Are there any canonical examples of Rigs, where the cancellative elements are not multiplicatively closed.
 
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  • #2
As there may not be any connection between + and . of R, Can(R) need not be closed.
 
  • #3
But there is a connection: distributivity.
 
  • #4
Consider those r in Can(R) which satisfy the additional condition that ra=rb implies a=b. They form a semiring.
 
  • #5

Thank you for providing a clear and detailed definition of a cancellative set in a semiring. I am always interested in exploring different mathematical structures and their properties.

To answer your question, yes, there are indeed examples of semirings where the cancellative elements are not multiplicatively closed. One such example is the semiring of non-negative integers with addition and multiplication as the operations. In this semiring, the set of cancellative elements is {0} since any non-zero element can be cancelled by multiplying it with 0. However, 0 is not a multiplicatively closed element as 0 multiplied by any non-zero element is still 0.

Another example is the semiring of polynomials over a field with addition and multiplication as the operations. In this semiring, the set of cancellative elements is the set of non-zero polynomials. However, this set is not multiplicatively closed as the product of two non-zero polynomials can still result in a zero polynomial.

In general, it can be shown that in any semiring, the set of cancellative elements is a submonoid of the additive monoid. However, it may not be multiplicatively closed. This is because the property of being multiplicatively closed depends on the specific operations defined in the semiring and may not hold for all elements.

I hope this response has provided some insight into the concept of cancellative sets in semirings and their properties. Thank you for bringing up this interesting topic for discussion.
 

1. What is a cancellative set in a semiring?

A cancellative set in a semiring is a subset of the semiring in which any element that can be canceled out on both sides of the multiplication operation. In other words, if a and b are in the set and ab = ac, then b = c. This property is also known as the cancellation property.

2. What is a semiring?

A semiring is an algebraic structure that is similar to a ring, but without the requirement for additive inverse elements. It consists of a set of elements, along with two binary operations: addition and multiplication. The set must follow the properties of associativity, commutativity, and distributivity for both operations.

3. Why is it important for a semiring to be multiplicatively closed?

A semiring must be multiplicatively closed in order for it to be considered a valid algebraic structure. This means that the result of multiplying any two elements in the set must also be in the set. If a semiring is not multiplicatively closed, it may not follow the properties required for a semiring, such as distributivity.

4. Can a set be cancellative in a semiring if it is not multiplicatively closed?

Yes, a set can be cancellative in a semiring even if it is not multiplicatively closed. The cancellation property only requires that the elements in the set can be canceled out on both sides of the multiplication operation. However, if the semiring is not multiplicatively closed, it may not follow other important properties, such as distributivity.

5. What are some examples of semirings that are not multiplicatively closed?

Some examples of semirings that are not multiplicatively closed include the set of positive real numbers under addition and multiplication, the set of non-negative integers under addition and multiplication, and the set of all integers under addition and multiplication.

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