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.
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.