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Wagner congruence
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(Definition)
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Let
 be the binary relation on the free semigroup with involution
 defined by
The Wagner congruence on  is the congruence  generated by
 , i.e.
 .
A well known result of inverse semigroups theory says that the quotient
is an inverse semigroup. Moreover
 is the Free Inverse Semigroup on  , in the sense that it resolve the following universal mapping problem: given an inverse semigroup  and a map
 , a unique inverse semigroups homomorphism
 exists such that the following diagram commutes:
where
 is the projection to the quotient, i.e.
![$ \iota(x)=[x]_{\rho_X}$](http://images.planetmath.org:8080/cache/objects/8272/l2h/img16.png "$ \iota(x)=[x]_{\rho_X}$") . It is well known from universal algebra that
 is unique up to isomorphisms.
In analogous way, using the free monoid with involution
 instead of the free semigroup with involution
 , we obtain the inverse monoid
that is the Free Inverse Monoid on  .
- 1
- N. Petrich, Inverse Semigroups, Wiley, New York, 1984.
- 2
- V.V. Wagner, Generalized Groups, Dokl. Akad. Nauk SSSR 84 (1952), 1119-1122.
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"Wagner congruence" is owned by Mazzu.
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(view preamble)
| Also defines: |
Wagner congruence, free inverse semigroup, free inverse monoid |
| Keywords: |
Inverse Semigroups, Free Object |
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Cross-references: monoid, inverse, free monoid with involution, isomorphisms, universal algebra, projection, homomorphism, mapping, universal, quotient, theory, inverse semigroups, generated by, congruence, free semigroup with involution, binary relation
There are 3 references to this entry.
This is version 12 of Wagner congruence, born on 2006-08-21, modified 2006-08-25.
Object id is 8272, canonical name is WagnerCongruence.
Accessed 1291 times total.
Classification:
| AMS MSC: | 20M18 (Group theory and generalizations :: Semigroups :: Inverse semigroups) | | | 20M05 (Group theory and generalizations :: Semigroups :: Free semigroups, generators and relations, word problems) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & X \ar[r]^{\iota} \ar[d]_{\Phi} & \mathrm{FIS}(X) \ar[dl]^{\overline{\Phi}} \ & S & } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/8272/l2h/img14.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & X \ar[r]^{\iota} \ar[d]_{\Phi} & \mathrm{FIS}(X) \ar[dl]^{\overline{\Phi}} \ & S & } } \end{xy}$")
