syntactic congruence

Let S be a semigroupPlanetmathPlanetmath and let XS. The relationMathworldPlanetmathPlanetmathPlanetmath

s1Xs2iffl,rS(ls1rXiffls2rX) (1)

is called the syntactic congruence of X. The quotient S/X is called the syntactic semigroup of X, and the natural morphismMathworldPlanetmathPlanetmath ϕ:SS/X is called the syntactic morphism of X. If S is a monoid, then S/X is also a monoid, called the syntactic monoid of X.

As an example, if S=(,+) and X={nkn=3k}, then mXn if mmod3=nmod3, and the syntactic monoid is isomorphicPlanetmathPlanetmathPlanetmath to the cyclic groupMathworldPlanetmath of order three.

It is straightforward that X is an equivalence relationMathworldPlanetmath and X is union of classes of X. To prove that it is a congruencePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, let s1,s2,t1,t2S satisfy s1Xs2 and t1Xt2. Let l,rS be arbitrary. Then ls1t1rX iff ls2t1rX because s1Xs2, and ls2t1rX iff ls2t2rX because t1Xt2. Then s1t1Xs2t2 since l and r are arbitrary.

The syntactic congruence is both left- and right-invariant, i.e., if s1Xs2, then ts1Xts2 and s1tXs2t for any t.

The syntactic congruence is maximal in the following sense:

  • if χ is a congruence over S and X is union of classes of χ,

  • then sχt implies sXt.

In fact, let l,rS: since sχt and χ is a congruence, lsrχltr. However, X is union of classes of χ, therefore lsr and ltr are either both in X or both outside X. This is true for all l,rS, thus sXt.

Title syntactic congruence
Canonical name SyntacticCongruence
Date of creation 2013-03-22 18:52:08
Last modified on 2013-03-22 18:52:08
Owner Ziosilvio (18733)
Last modified by Ziosilvio (18733)
Numerical id 6
Author Ziosilvio (18733)
Entry type Definition
Classification msc 68Q70
Classification msc 20M35
Defines syntactic semigroup
Defines syntactic monoid
Defines syntactic morphism
Defines maximality property of syntactic congruence