abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, a branch of mathematics, a maximal semilattice quotient is a

commutative monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...

derived from another commutative monoid by making certain elements

equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equivale ...

to each other. Every commutative monoid can be endowed with its ''algebraic''

preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...

ing ≤ . By definition, ''x≤ 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 In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

\asymp

is a

monoid congruence In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...

of ''M'', and the quotient monoid

M/

is the ''maximal semilattice quotient'' of ''M''. This terminology can be explained by the fact that the canonical projection ''p'' from ''M'' onto

M/

is universal among all monoid homomorphisms from ''M'' to a (∨,0)-

semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has ...

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

g\colon M/\to S

such that ''f=gp''. If ''M'' is a

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'' ...

, then

M/

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. Lattice theory {{algebra-stub