Maximal semilattice quotient

In abstract algebra, a branch of mathematics, a maximal semilattice quotient is a commutative monoid derived from another commutative monoid by making certain elements equivalent to each other.
Every commutative monoid can be endowed with its "algebraic" preordering ≤ . By definition, "x≤ y" holds, if there exists "z" such that "x+z=y". Further, for "x, y" in "M", let formula_1 hold, if there exists a positive integer "n" such that "x≤ ny", and let formula_2 hold, if formula_1 and formula_4. The binary relation formula_5 is a monoid congruence of "M", and the quotient monoid formula_6 is the "maximal semilattice quotient" of "M".
This terminology can be explained by the fact that the canonical projection "p" from "M" onto formula_6 is universal among all monoid homomorphisms from "M" to a (∨,0)-semilattice, that is, for any (∨,0)-semilattice "S" and any monoid homomorphism "f: M→ S", there exists a unique (∨,0)-homomorphism formula_8 such that "f=gp".
If "M" is a refinement monoid, then formula_6 is a distributive semilattice.
References.
A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups. Vol. I. Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1961. xv+224 p.