Incline

An incline has two associative commutative operations $+$ and multiplication, with multiplication distributing over addition, idempotent addition and absorption $x+xy=x$. Neither operation is required here to have an identity. Its natural order is $x\le y⟺x+y=y$, making addition a join. Multiplication is order-preserving, and products lie below both factors. A finitely generated incline is well-quasi-ordered by its reversed natural order, via the reverse-inclusion well-quasi-ordering of monomial ideals applied to exponent supports.

Expand every term in finitely many generators as a finite nonempty sum of monomials, and associate its exponent support with the upward-closed subset of ${\mathbb{N}}^r$ that it generates. The reverse-inclusion well-quasi-ordering of monomial ideals supplies an earlier support upset containing a later one. Every later monomial is then a multiple of an earlier monomial, hence is absorbed by it under $a+ab=a$. The earlier sum absorbs the later sum. Thus every infinite term sequence has an earlier element preceding a later one in the reverse natural order. Reflexivity, antisymmetry and transitivity follow directly from commutative idempotent addition.