Bicyclic subsemigroups in amalgams of finite inverse semigroups

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2 whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max{|S1|,|S 2|}. Moreover we consider amalgams of finite inverse semigroups respecting the J-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the R-classes to be finite. © 2010 World Scientific Publishing Company.