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In [[abstract algebra]], a branch of [[mathematics]], a '''maximal semilattice quotient''' is a [[commutative monoid]] derived from another [[commutative monoid]] by making certain elements [[equivalence relation|equivalent]] to each other. | |||
Every [[Monoid|commutative monoid]] can be endowed with its ''algebraic'' [[preorder]]ing ≤ . By definition, ''x≤ y'' holds, if there exists ''z'' such that ''x+z=y''. Further, for ''x, y'' in ''M'', let <math>x\propto y</math> hold, if there exists a positive integer ''n'' such that ''x≤ ny'', and let <math>x\asymp y</math> hold, if <math>x\propto y</math> and <math>y\propto x</math>. The [[binary relation]] <math>\asymp</math> is a [[Congruence relation|monoid congruence]] of ''M'', and the quotient monoid <math>M/{\asymp}</math> is the ''maximal semilattice quotient'' of ''M''. | |||
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This terminology can be explained by the fact that the canonical projection ''p'' from ''M'' onto <math>M/{\asymp}</math> 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 <math>g\colon M/{\asymp}\to S</math> such that ''f=gp''. | |||
If ''M'' is a [[refinement monoid]], then <math>M/{\asymp}</math> is a [[Distributivity (order theory)|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. | |||
{{DEFAULTSORT:Maximal Semilattice Quotient}} | |||
[[Category:Lattice theory]] | |||
{{algebra-stub}} | |||
Revision as of 15:39, 22 June 2013
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 (∨,0)-semilattice, 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, then 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.