Inverse semigroup amalgams with a lower bounded core

Abstract

We consider inverse semigroup amalgams [S1,S2;U] such that for any u ∈ U and e ∈ E(Si) with u ≥ e in Si, where i ∈ {1,2}, there exists f ∈ E(U) with u ≥ f ≥ e in Si; we say that U is lower bounded in S1 and S2. We construct and describe the Schutzenberger automata of S1∗U S2 and give conditions for decidable word problem. The homomorphisms of the Schutzenberger graphs of S1∗U S2 are studied and conditions are given for S1∗U S2 to be completely semisimple. In the case when S1 and S2 have decidable word problems and U is finite, we show that S1∗U S2 has decidable word problem.