|
|
| Line 1: |
Line 1: |
| {{Expert-subject|date=March 2009}}
| | Hi there, I am Yoshiko Villareal but I never really liked that name. His working day occupation is a monetary officer but he ideas on altering it. Some time ago I chose to live in Arizona but I need to move for my family. The thing I adore most bottle tops gathering and now I have time to take on new issues.<br><br>Here is my web page [http://bjjoutlet.com/UserProfile/tabid/43/userId/20679/Default.aspx http://bjjoutlet.com] |
| | |
| 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''.
| |
| <br />
| |
| 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}}
| |
Latest revision as of 10:44, 24 July 2014
Hi there, I am Yoshiko Villareal but I never really liked that name. His working day occupation is a monetary officer but he ideas on altering it. Some time ago I chose to live in Arizona but I need to move for my family. The thing I adore most bottle tops gathering and now I have time to take on new issues.
Here is my web page http://bjjoutlet.com