free semigroup with involution

Let X,X be two disjoint sets in bijectiveMathworldPlanetmathPlanetmath correspondence given by the map :XX. Denote by Y=XX (here we use instead of to remind that the union is actually a disjoint unionMathworldPlanetmathPlanetmath) and by Y+ the free semigroupMathworldPlanetmath on Y. We can extend the map to an involution :Y+Y+ on Y+ in the following way: given wY+, we have w=w1w2wk for some letters wiY; then we define


It is easily verified that this is the unique way to extend to an involution on Y. Thus, the semigroupPlanetmathPlanetmath (XX)+ with the involution is a semigroup with involution. Moreover, it is the free semigroup with involution on X, in the sense that it solves the following universal problem: given a semigroup with involution S and a map Φ:XS, a semigroup homomorphism Φ¯:(XX)+S exists such that the following diagram commutes:


where ι:X(XX)+ is the inclusion mapMathworldPlanetmath. It is well known from universal algebraMathworldPlanetmathPlanetmath that (XX)+ is unique up to isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

If we use Y* instead of Y+, where Y*=Y+{ε} and ε is the empty wordPlanetmathPlanetmath (i.e. the identityPlanetmathPlanetmathPlanetmathPlanetmath of the monoid Y*), we obtain a monoid with involution (XX)* that is the free monoid with involution on X.

Title free semigroup with involution
Canonical name FreeSemigroupWithInvolution
Date of creation 2013-03-22 16:11:30
Last modified on 2013-03-22 16:11:30
Owner Mazzu (14365)
Last modified by Mazzu (14365)
Numerical id 8
Author Mazzu (14365)
Entry type Example
Classification msc 20M10
Defines free semigroup with involution
Defines free monoid with involution