J- Simple restriction ω-semigroups

In 1985 Asibong-Ibe [1] considered the �*-bisimple ample ω-semigroups
and proved that they are isomorphic to certain generalised Bruck-Reilly extensions
BR(M;θ) of a cancellative monoid M where θ is a morphism.
In 1993 he also proved in [2] that a similar structure theorem holds for
J-simple ample ω-semigroups. Recently this result has been generalised in [16] for ~-bisimple restriction ω-semigroups.
Our objective is to study generalised Green's relations and restriction semigroups, together with their properties. Then we show that the analogue of a similar structure theorem for simple inverse ω-semigroups that was proved in [13] still holds if we replace simple inverse ω-semigroups by J-simple restriction ω-semigroups. The theory developed here closely parallels
that in [13] and [2].