addition replaced by minimum and multiplication by ordinary addition. The unit for is (the “zero”), and the unit for is . It is a semiring rather than a ring, because is idempotent () and has no inverse, so subtraction is absent. The max-plus convention is the order-reversed twin. Both are used, and this wiki takes for shapes so that “larger” reads as “outermost”.
A tropical monomial is the affine function , and a tropical polynomial is therefore a piecewise-linear, concave function (convex under ). Polynomials become minima of affine pieces; their non-smooth locus is where two pieces tie.
The semiring is the dequantisation ( limit) of under . In that limit sums of exponentials collapse to their single dominant term, giving , and products become sums. Tropical objects are therefore the “log-limit” skeletons of ordinary algebraic ones; see tropical-variety.