Orthogonal completions of the polycyclic monoids

The polycyclic monoids Pn and the Thompson groups Vn,1

We construct what we call the strong orthogonal completion Cn of the polycyclic monoid Pn on n generators. The inverse monoid Cn is congruence free and its group of units is the Thompson group Vn,1. Copies of Cn can be constructed from partitions of sets into n blocks each block having the same cardinality as the underlying set. 2000 AMS Subject Classification: 20M18, 20E32.

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We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 (that is...

Primitive representations of the polycyclic monoids and branching function systems

We generalise the group theoretic notion of a primitive permutation representation to inverse monoids and so obtain a notion of a primitive representation by partial permutations. Such representations are shown to be determined by what we call essentially maximal proper closed inverse submonoids. Such submonoids in the case of the polycyclic inverse monoids (also known as Cuntz inverse semigrou...

Quantales as completions of ordered monoids - Revised semantics for Intuitionistic Linear Logic

Primitive partial permutation representations of the polycyclic monoids and branching function systems

We characterise the maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse semigroups, and so determine all their primitive partial permutation representations. We relate our results to the work of Kawamura on certain kinds of representations of the Cuntz C∗-algebras and to the branching function systems of Bratteli and Jorgensen.