 Solution

Suppose the incline is generated by $x_1,\dots ,x_r$. Repeated distributivity, followed by commutativity and additive idempotence, expresses every element as a finite nonempty sumNo multiplicative identity is assumed: a zero exponent means that the corresponding generator is omitted, and every displayed monomial has at least one factor. Different supports may represent the same element; uniqueness is unnecessary.

Associate to $A$ its upward-closed exponent setIf $U_A\supseteq U_B$, each monomial of $f_B$ is either equal to, or a multiplicative extension of, a monomial of $f_A$. It is absorbed by that earlier monomial, and hence by the full sum $f_A$. Adding the terms of $f_B$ one at a time givesFor equal exponent vectors the first absorption is additive idempotence; for strictly larger vectors it is the incline's multiplicative absorption. Thus the issue is to prove that exponent upsets are well-quasi-ordered by reverse inclusion, not merely to apply Dickson's lemma to individual exponents.