Ternary rings of operators arising from inverse semigroups

We are interested in properties, especially injectivity (in the sense of category theory), of the ternary rings of operators generated by certain subsets of an inverse semigroup via the regular representation. We determine all subsets of the extended bicyclic semigroup which are closed under the triple product $x{y}^{\ast }z$ (called semiheaps) and show that the weakly closed ternary rings of operators generated by them (W*-TROs) are injective operator spaces.