In light of the two leading candidates for the title of "tropical variety", we need to distinguish two competing (cooperating?) notions of tropical lines: 1) bend loci defined by linear tropical polynomials; and 2) congruence varieties defined by linear tropical equations.

Linear bend loci

The bendy option for tropical lines is the bend locus of a linear tropical polynomial. Focusing on planar lines, these are the bend loci of polynomials of the form $f(x,y)=(a\odot x)\oplus (b\odot y)\oplus c$, where $a,b,c\in \mathbf{T}$ (and at least one of $a,b$ is not equal to $0_{\mathbf{T}}=-\mathrm{\infty }$.) All such bend loci have the same combinatorial shape, namely that of three rays coming (one horizontal in the negative $x$-direction, one vertical in the negative $y$-direction, and one with slope one in the positive $x,y$-direction) meeting at a single vertex:

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The location of the vertex is easily deduced from the linear equation, and vice versa.

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Linear congruence varieties

The congruence option for tropical lines is the congruence variety of a quadratic linear equation, i.e., an equation of the form $f(x,y)=g(x,y)$, involving two tropical polynomials of degree at most one (and at least one term of degree one). These congruence varieties share many similar features with the bend loci above, but also have the unique possibility of including some two-dimensional regions in the plane.

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