On omitted value for transcendental semigroups

In this paper, we define omitted value for transcendental semigroups and show that in a transcendental semigroup there can be at most one omitted value. Several examples of transcendental semigroups having omitted value are given. We define Baker wandering domain for transcendental semigroups. For a transcendental semigroup having a Baker wandering domain W as well as an omitted value a, we prove that a lies in the Julia set. We show that if two transcendental semigroups G and $$G^{\prime }$$ G ′ are conjugate under a map $$\psi $$ ψ and G has omitted value $$z_0$$ z 0 then $$G^{\prime }$$ G ′ has omitted value $$\psi (z_0)$$ ψ ( z 0 ) . We show that if $$G=\; $$ G = such that g is conjugate of f under the map $$\psi $$ ψ and f has omitted value $$z_0$$ z 0 then $$O(G)=\{z_0\}$$ O ( G ) = { z 0 } if and only if $$\psi (z_0)\,=\,z_0$$ ψ ( z 0 ) = z 0 . Finally, we conclude this paper by giving some problems for future research.