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Following my grade school education, I will call the set $\mathbb{N}_0\equiv\left\{0,1,2,\dots\right\}$ the whole numbers, and the set $\mathbb{N}\equiv\left\{1,2,3,\dots\right\}$ the natural numbers. The following is a list of what I believe to be the essential differences between the properties of $\left<\mathbb{N},+,\times\right>$ versus $\left<\mathbb{N}_0,+,\times\right>$:

$$\begin{array}{ccccc} \text{Law} & \text{op} & \text{Name} & \mathbb{N} & \mathbb{N}_{0}\\ a+\left(b+c\right)=\left(a+b\right)+c & \left[+\right] & \text{Associative} & \text{T} & \text{T}\\ a+b=b+a & \left[+\right] & \text{Commutative} & \text{T} & \text{T}\\ a+c=b+c\implies a=b & \left[+\right] & \text{Cancellative} & \text{T} & \text{T}\\ a+0=a & \left[+\right] & \text{Identative} & \text{F} & \text{T}\\ 0+0=0 & \left[+\right] & 0\text{--Idempotent} & \text{F} & \text{T}\\ a\times\left(b\times c\right)=\left(a\times b\right)\times c & \left[\times\right] & \text{Associative} & \text{T} & \text{T}\\ a\times b=b\times a & \left[\times\right] & \text{Commutative} & \text{T} & \text{T}\\ a\times c=b\times c\implies a=b & \left[\times\right] & \text{Cancellative} & \text{T} & \mathbb{N}\\ a\times1=a & \left[\times\right] & \text{Identative} & \text{T} & \text{T}\\ a\times\left(b+c\right)=a\times b+a\times c & \left[\times\right] & \text{Distributive} & \text{T} & \text{T}\\ 1\times1=1 & \left[\times\right] & 1\text{--Idempotent} & \text{T} & \text{T}\\ 0\times0=0 & \left[\times\right] & 0\text{--Idempotent} & \text{F} & \text{T}\\ 0\times a=0 & \left[\times\right] & 0\text{--Degeneracy} & \text{F} & \text{T} \end{array}$$

Thurston calls the additive algebraic structure $\left<\mathbb{N}_0,+\right>$ a hemigroup. This is also known as a cancellative, commutative monoid.

The additive algebraic structure $\left<\mathbb{N},+\right>$ is a cancellative, commutative semigroup. I have no special name for this structure.

The multiplicative structure $\left<\mathbb{N},\times\right>$ is a hemigroup, but $\left<\mathbb{N}_0,\times\right>$ fails to be a hemigroup because $a\times 0=b\times 0$ is degenerate. For a hemigroup the unique idempotent element can be shown to be the unique identity element. I find it modestly interesting that this applies to $\left<\mathbb{N}_0,+\right>$ and to $\left<\mathbb{N},\times\right>$ but not to the other two structures.

My question is this. Are there any special names given to the algebraic structures $\left<\mathbb{N},+,\times\right>$ determined by the natural numbers and $\left<\mathbb{N}_0,+,\times\right>$ determined by the whole numbers?

I am also interested in any corrections to or obvious omissions in my list.

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    $\begingroup$ You mean semigroup. Thurston is right. $\endgroup$ – Wuestenfux May 12 at 6:22
  • $\begingroup$ No. I mean hemigroup. archive.org/details/TheNumberSystem/page/n35 $\endgroup$ – Steven Thomas Hatton May 13 at 10:44
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    $\begingroup$ @StevenThomasHatton I would advise not to use this term: it is highly non-standard, and apparently seems to be used for other, totally unrelated, structures: link.springer.com/article/10.1007/BF02189348 $\endgroup$ – lisyarus May 13 at 10:50
  • $\begingroup$ I've seen the source you referenced. But, so long as we are clear about our context, I don't see a compelling reason not to overload the definition. I'm also considering calling a cancellative commutative semigroup a natural semigroup. Then, perhaps I should call Thurston's hemigroup a whole semigroup. Some m.se participants are actively disinterested in the comparison between $\mathbb{N}$ and $\mathbb{N}_0$. My previous request for references on the topic was voted down and out. And I haven't found much in the way of meaningful discussion elsewhere. I guess it's up to me. $\endgroup$ – Steven Thomas Hatton May 13 at 11:02
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Let me use a different notation. Let $\mathbb{N} = \{ 0, 1, \dotsm \}$ be the set of all natural numbers and let $\mathbb{N}_{> 0} = \mathbb{N} - \{0\} = \{ 1, 2, \dotsm \}$ be the set of positive integers.

Then $(\mathbb{N}, +)$ and $(\mathbb{N}_{> 0}, +)$ are respectively the free monoid and the free semigroup on one generator. Moreover, $(\mathbb{N}, +, \times)$ is usually known as the semiring of natural numbers.

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