Free completely J (.)-simple Semigroups

A semigroup is called completely J (.) -simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J (.)-simple semigroups form a quasivarity. Moreover, the construction of free completely J (.)-simple semigroups is given. It is found that a free completely J (.)-simple semigroup is just a free completely J*-simple semigroup and also a full subsemigroup of some completely simple semigroups.