The concept of annihilator ideal is introduced in an Almost Semi lattice (ASL) L with 0. It is proved that the set of all annihilator ideals of an ASL L with 0 forms a complete Boolean algebra. The concept of annihilator preserving homomorphism is introduced in an ASL L with 0. A sufficient condition for a homomorphism to be annihilator preserving is derived. Finally, it is proved that the homomorphism image and the inverse image of an annihilator ideal are again annihilator ideals.