In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a maximal semilattice quotient is a commutative monoid derived from another commutative monoid by making certain elements equivalent

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

 to each other.

Every commutative monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

 can be endowed with its algebraic preorder

Preorder

In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...

ing ≤ . 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

Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

  is a monoid congruence

Congruence relation

In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

 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 (∨,0)-semilattice

Semilattice

In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...

, that is, for any (∨,0)-semilattice S and any monoid homomorphism f: M→ S, there exists a unique (∨,0)-homomorphism such that f=gp.

If M is a refinement monoid

Refinement monoid

In mathematics, a refinement monoid is a commutative monoid M such that for any elements a0, a1, b0, b1 of M such that a0+a1=b0+b1, there are elements c00, c01, c10, c11 of M such that a0=c00+c01, a1=c10+c11, b0=c00+c10, and b1=c01+c11.A commutative monoid M is conical, if x+y=0 implies that x=y=0,...

, then is a distributive semilattice

Distributivity (order theory)

In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima...

.