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.
hold, if there exists a positive integer n such that x≤ ny, and let
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 (∨,0)-semilattice, that is, for any (∨,0)-semilattice S and any monoid homomorphism f: M→ S, there exists a unique (∨,0)-homomorphism
If M is a refinement monoid, then
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