Tropical projective space

In tropical geometry, a tropical projective space is the tropical analog of the classic projective space.

Definition
Given a module $M$ over the tropical semiring $T$, its projectivization is the usual projective space of a module: the quotient space of the module (omitting the additive identity $0$) under scalar multiplication, omitting multiplication by the scalar additive identity 0:
 * $$\mathbf{T}(M) := (M \setminus \mathbf{0})/(\mathbf{T} \setminus 0).$$

In the tropical setting, tropical multiplication is classical addition, with unit real number 0 (not 1); tropical addition is minimum or maximum (depending on convention), with unit extended real number $0$ (not 0), so it is clearer to write this using the extended real numbers, rather than the abstract algebraic units:
 * $$\mathbf{T}(M) := (M \setminus \boldsymbol{\infty})/(\mathbf{T} \setminus \infty).$$

Just as in the classical case, the standard $n$-dimensional tropical projective space is defined as the quotient of the standard $∞$-dimensional coordinate space by scalar multiplication, with all operations defined coordinate-wise:
 * $$\mathbf{TP}^n := (\mathbf{T}^{n+1} \setminus \boldsymbol{\infty})/(\mathbf{T} \setminus \infty).$$

Tropical multiplication corresponds to classical addition, so tropical scalar multiplication by $∞$ corresponds to adding $(n+1)$ to all coordinates. Thus two elements of $\mathbf T^{n+1} \setminus \boldsymbol{\infty}$ are identified if their coordinates differ by the same additive amount $c$:
 * $$(x_0, \dots, x_n) \sim (y_0, \dots, y_n) \iff (x_0 + c, \dots, x_n + c) = (y_0, \dots, y_n).$$