Monoid generalizations of the Richard Thompson groups

The groups Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,1, and to inverse monoids, called Invk,1; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids Mk,1 have connections with circuit complexity (studied in another paper). Here we prove that Mk,1 and Invk,1 are congruence-simple for all k. Their Green relations J and D are characterized: Mk,1 and Invk,1 are J-0-simple, and they have k− 1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra Ok. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets. 1 Thompson-Higman monoids Since their introduction by Richard J. Thompson in the mid 1960s [26, 23, 27], the Thompson groups have had a great impact on infinite group theory. Graham Higman generalized the Thompson groups to an infinite family [17]. These groups and some of their subgroups have appeared in many contexts and have been widely studied; see for example [9, 5, 12, 7, 14, 15, 6, 8, 20]. The definition of the Thompson-Higman groups lends itself easily to generalizations to inverse monoids and to more general monoids. These monoids are also generalizations of the finite symmetric monoids (of all functions on a set), and this leads to connections with circuit complexity; more details on this appear in [1, 2, 4]. By definition the Thompson-Higman group Gk,1 consists of all maximally extended isomorphisms between finitely generated essential right ideals of A∗, where A is an alphabet of cardinality k. The multiplication is defined to be composition followed by maximal extension: for any φ,ψ ∈ Gk,1, we have φ · ψ = max(φ ◦ ψ). Every element φ ∈ Gk,1 can also be given by a bijection φ : P → Q where P,Q ⊂ A∗ are two finite maximal prefix codes over A; this bijection can be described concretely by a finite function table. For a detailed definition according to this approach, see [3] (which is also similar to [25], but with a different terminology); moreover, Subsection 1.1 gives all the needed definitions. It is natural to generalize the maximally extended isomorphisms between finitely generated essential right ideals of A∗ to homomorphisms, and to drop the requirement that the right ideals be essential. It will turn out that this generalization leads to interesting monoids, or inverse monoids, which we call Thompson-Higman monoids. Our generalization of the Thompson-Higman groups to monoids will also generalize the embedding of these groups into the Cuntz algebras [3, 24], which provides an additional motivation for our definition. Moreover, since these homomorphisms are close to being arbitrary finite string transformations, there is a connection between these monoids and combinational boolean circuits; the study of the connection between Thompson-Higman groups and circuits was started in ∗Supported by NSF grant CCR-0310793. The first version of this paper appeared in http://arxiv.org/abs/0704.0189 (2 April 2007).