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 hold, if there exists a positive integer n such that x≤ ny, and let hold, if and. The binary relation is a monoid congruence of M, and the quotient monoid is the maximal semilattice quotient of M.
This terminology can be explained by the fact that the canonical projection p from M onto is universal among all monoid homomorphisms from M to a -semilattice, that is, for any -semilattice S and any monoid homomorphism f: M→ S, there exists a unique -homomorphism such that f=gp.
If M is a refinement monoid, then is a distributive semilattice.
