HNN extensions with lower bounded inverse monoids
Paul Bennett, Tatiana B. Jajcayová · Communications in Algebra · 2022
We consider HNN extensions S*=[S;U1,U2;ϕ] where U1 and U2 are inverse monoids of an inverse semigroup S such that, for any u∈Ui and e∈E(S) with u≥e in S, there exists f∈E(Ui) with u≥f≥e in S, for i∈{1,2}; we say that U1 and U2 are lower bounded in S. We construct and describe the Schützenberger automata of S* and give conditions for S* to have decidable word problem. Homomorphisms of the Schützenberger graphs of S* are studied and conditions are given for S* to be completely semisimple. When S has decidable word problem and U1 and U2 are finite, we show that S* has decidable word problem. The class of HNN extensions considered here is surprisingly useful and generalizes the class introduced by Jajcayová. A future paper intends to show that any HNN extension of an inverse semigroup can be embedded into an HNN extension where the subsemigroups are lower bounded.