Strong representations of the polycyclic inverse monoids: Cycles and atoms

AbstractThis paper was inspired by a monograph by Bratteli and Jorgensen, and the work of Kawamura. We introduce two new semigroups: a wide inverse submonoid of the polycyclic inverse monoid, called the gauge inverse monoid, and a Zappa-Szép product of an arbitrary free monoid with the free monoid on one generator. Both these monoids play an important role in studying arbitrary, not necessarily transitive, strong actions of polycyclic inverse monoids. As a special case of such actions, we obtain some new results concerning the strong actions of P2 on ℤ determined by the choice of one positive odd number. We explain the role played by Lyndon words in characterising these repesentations and show that the structure of the representation can be explained by studying the binary representations of the numbers $$\frac{1} {p},\frac{2} {p}, \ldots \frac{{p - 1}} {p}$$. We also raise some questions about strong representations of the polycyclic monoids on free abelian groups.

[1]  David R. Pitts,et al.  Invariant Subspaces and Hyper‐Reflexivity for Free Semigroup Algebras , 1999 .

[2]  Mark V. Lawson,et al.  The Polycyclic Monoids P n and the Thompson Groups V n,1 , 2007 .

[3]  Christian Choffrut,et al.  Combinatorics of Words , 1997, Handbook of Formal Languages.

[4]  Palle E.T. Jorgensen,et al.  Iterated Function Systems and Permutation Representations of the Cuntz Algebra , 1996 .

[5]  J. Renault.  A Groupoid Approach to C*-Algebras , 1980 .

[6]  J. Howie.  Fundamentals of semigroup theory , 1995 .

[7]  KATSUNORI KAWAMURA.  Branching Laws for Polynomial Endomorphisms of Cuntz Algebras Arising from Permutations , 2006 .

[8]  Volodymyr Nekrashevych,et al.  Self-Similar Groups , 2005, 2304.11232.

[9]  Joachim Cuntz,et al.  SimpleC*-algebra generated by isometries , 1977 .

[10]  John Meakin,et al.  Congruences on free monoids and submonoids of polycyclic monoids , 1993 .

[11]  S. Lang.  Algebraic Number Theory , 1971 .

[12]  Derek J. S. Robinson,et al.  On Groups that are Isomorphic with Every Subgroup of Finite Index and their Topology , 1998 .

[13]  Katsunori Kawamura.  Polynomial Endomorphisms of the Cuntz Algebras Arising from Permutations: I – General Theory , 2005 .

[14]  Mark V. Lawson.  Primitive partial permutation representations of the polycyclic monoids and branching function systems , 2009, Period. Math. Hung..

[15]  Mark V. Lawson,et al.  Orthogonal Completions of the Polycyclic Monoids , 2007 .

[16]  M. Lothaire,et al.  Combinatorics on words: Frontmatter , 1997 .

[17]  Dan Kalman,et al.  Fractions with Cycling Digit Patterns , 1996 .

[18]  I. Stewart,et al.  Algebraic Number Theory , 1992, All the Math You Missed.

[19]  Mark V Lawson,et al.  A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY , 2009, Journal of the Australian Mathematical Society.

[20]  M. Lothaire.  Combinatorics on words: Bibliography , 1997 .

[21]  Mark V. Lawson.  A correspondence between a class of monoids and self-similar group actions I , 2008 .

[22]  J. Berstel,et al.  Theory of codes , 1985 .

[23]  A. Paterson,et al.  Groupoids, Inverse Semigroups, and their Operator Algebras , 1998 .