bicyclic semigroup


The bicyclic semigroup 𝒞⁢(p,q) is the monoid generated by {p,q} with the single relation p⁢q=1.

The elements of 𝒞⁢(p,q) are all words of the form qn⁢pm for m,n≥0 (with the understanding p0=q0=1). These words are multiplied as follows:

qn⁢pm⁢qk⁢pl={qn+k-m⁢plif ⁢m≤k,qn⁢pl+m-kif ⁢m≥k.

It is apparent that 𝒞⁢(p,q) is simple, for if qn⁢pm is an element of 𝒞⁢(p,q), then 1=pn⁢(qn⁢pm)⁢qm and so S1⁢qn⁢pm⁢S1=S.

It is also easy to see that 𝒞⁢(p,q) is an inverse semigroup: the element qn⁢pm has inverseMathworldPlanetmathPlanetmathPlanetmath qm⁢pn.

It is useful to picture some further properties of 𝒞⁢(p,q) by arranging the elements in a table:

1pp2p3p4…qq⁢pq⁢p2q⁢p3q⁢p4…q2q2⁢pq2⁢p2q2⁢p3q2⁢p4…q3q3⁢pq3⁢p2q3⁢p3q3⁢p4…q4q4⁢pq4⁢p2q4⁢p3q4⁢p4…⋮⋮⋮⋮⋮⋱

Then the elements below any horizontal line drawn through this table form a right idealMathworldPlanetmathPlanetmath and the elements to the right of any vertical line form a left ideal. Further, the elements on the diagonal are all idempotentsPlanetmathPlanetmath and their standard ordering is

1>q⁢p>q2⁢p2>q3⁢p3>⋯.
Title bicyclic semigroup
Canonical name BicyclicSemigroup
Date of creation 2013-03-22 13:09:57
Last modified on 2013-03-22 13:09:57
Owner mclase (549)
Last modified by mclase (549)
Numerical id 8
Author mclase (549)
Entry type Definition
Classification msc 20M99
Synonym bicyclic monoid
Defines bicyclic