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&le; y holds, if there exists z such that x+z=y. Further, for x, y in M, let $$x\propto y$$ hold, if there exists a positive integer n such that x≤ ny, and let $$x\asymp y$$ hold, if $$x\propto y$$ and $$y\propto x$$. The binary relation $$\asymp$$ is a monoid congruence of M, and the quotient monoid $$M/{\asymp}$$ is the maximal semilattice quotient of M.

This terminology can be explained by the fact that the canonical projection p from M onto $$M/{\asymp}$$ is universal among all monoid homomorphisms from M to a (&or;,0)-semilattice, that is, for any (&or;,0)-semilattice S and any monoid homomorphism f: M→ S, there exists a unique (&or;,0)-homomorphism $$g\colon M/{\asymp}\to S$$ such that f=gp.

If M is a refinement monoid, then $$M/{\asymp}$$ is a distributive semilattice.