diff --git a/src/Singleton.jl b/src/Singleton.jl
index 112e0de6ed..76d3b4f4f4 100644
--- a/src/Singleton.jl
+++ b/src/Singleton.jl
@@ -3,7 +3,7 @@ import Base: ∈, ⊆
 export Singleton
 
 """
-    Singleton{N<:Real} <: LazySet
+    Singleton{N<:Real} <: AbstractSingleton{N}
 
 Type that represents a singleton, that is, a set with a unique element.
 
@@ -11,95 +11,45 @@ Type that represents a singleton, that is, a set with a unique element.
 
 - `element` -- the only element of the set
 """
-struct Singleton{N<:Real} <: LazySet
+struct Singleton{N<:Real} <: AbstractSingleton{N}
     element::Vector{N}
 end
 
-"""
-    dim(S::Singleton)::Int
-
-Return the dimension of a singleton.
 
-### Input
+# --- AbstractSingleton interface functions ---
 
-- `S` -- singleton
 
-### Output
-
-The ambient dimension of the singleton.
 """
-function dim(S::Singleton)::Int
-    return length(S.element)
-end
+    element(S::Singleton{N})::Vector{N} where {N<:Real}
 
-"""
-    σ(d::AbstractVector{<:Real}, S::LazySets.Singleton{N})::Vector{N} where {N<:Real}
-
-Return the support vector of a singleton.
+Return the element of a singleton.
 
 ### Input
 
-- `d` -- direction
-- `B` -- singleton
+- `S` -- singleton
 
 ### Output
 
-The support vector, which is the singleton's vector itself, irrespective of the
-given direction.
+The element of the singleton.
 """
-function σ(d::AbstractVector{<:Real},
-           S::LazySets.Singleton{N})::Vector{N} where {N<:Real}
+function element(S::Singleton{N})::Vector{N} where {N<:Real}
     return S.element
 end
 
 """
-    ∈(x::AbstractVector{N}, S::Singleton{N})::Bool where {N<:Real}
+    element(S::Singleton{N}, i::Int)::N where {N<:Real}
 
-Check whether a given point is contained in a singleton.
+Return the i-th entry of the element of a singleton.
 
 ### Input
 
-- `x` -- point/vector
 - `S` -- singleton
+- `i` -- dimension
 
 ### Output
 
-`true` iff ``x ∈ S``.
-
-### Notes
-
-This implementation performs an exact comparison, which may be insufficient with
-floating point computations.
-
-### Examples
-
-```jldoctest
-julia> S = Singleton([1., 1.]);
-
-julia> ∈([0.9, 1.1], S)
-false
-julia> ∈([1.0, 1.0], S)
-true
-```
-"""
-function ∈(x::AbstractVector{N}, S::Singleton{N})::Bool where {N<:Real}
-    return x == S.element
-end
-
-"""
-    ⊆(S::Singleton, set::LazySet)::Bool
-
-Check whether a given singleton is contained in a convex set.
-
-### Input
-
-- `S`   -- singleton
-- `set` -- convex set
-
-### Output
-
-`true` iff ``S ⊆ \\text{set}``.
+The i-th entry of the element of the singleton.
 """
-function ⊆(S::Singleton, set::LazySet)::Bool
-    return ∈(S.element, set)
+function element(S::Singleton{N}, i::Int)::N where {N<:Real}
+    return S.element[i]
 end
diff --git a/src/ZeroSet.jl b/src/ZeroSet.jl
index 27cf160428..c19d68bfdc 100644
--- a/src/ZeroSet.jl
+++ b/src/ZeroSet.jl
@@ -3,7 +3,7 @@ import Base.∈
 export ZeroSet
 
 """
-    ZeroSet <: LazySet
+    ZeroSet{N<:Real} <: AbstractSingleton{N}
 
 Type that represents the zero set, i.e., the set that only contains the origin.
 
@@ -11,9 +11,54 @@ Type that represents the zero set, i.e., the set that only contains the origin.
 
 - `dim` -- the ambient dimension of this zero set
 """
-struct ZeroSet <: LazySet
+struct ZeroSet{N<:Real} <: AbstractSingleton{N}
     dim::Int
 end
+# type-less convenience constructor
+ZeroSet(dim::Int) = ZeroSet{Float64}(dim)
+
+
+# --- AbstractSingleton interface functions ---
+
+
+"""
+    element(S::ZeroSet{N})::Vector{N} where {N<:Real}
+
+Return the element of a zero set.
+
+### Input
+
+- `S` -- zero set
+
+### Output
+
+The element of the zero set, i.e., a zero vector.
+"""
+function element(S::ZeroSet{N})::Vector{N} where {N<:Real}
+    return zeros(N, S.dim)
+end
+
+"""
+    element(S::ZeroSet{N}, ::Int)::N where {N<:Real}
+
+Return the i-th entry of the element of a zero set.
+
+### Input
+
+- `S` -- zero set
+- `i` -- dimension
+
+### Output
+
+The i-th entry of the element of the zero set, i.e., 0.
+"""
+function element(S::ZeroSet{N}, ::Int)::N where {N<:Real}
+    return zero(N)
+end
+
+
+# --- LazySet interface functions ---
+
 
 """
     dim(Z::ZeroSet)::Int
@@ -33,7 +78,7 @@ function dim(Z::ZeroSet)::Int
 end
 
 """
-    σ(d, Z)
+    σ(d::AbstractVector{N}, Z::ZeroSet)::Vector{N} where {N<:Real}
 
 Return the support vector of a zero set.
 
@@ -51,7 +96,7 @@ function σ(d::AbstractVector{N}, Z::ZeroSet)::Vector{N} where {N<:Real}
 end
 
 """
-    ∈(x::AbstractVector, Z::ZeroSet)::Bool
+    ∈(x::AbstractVector{N}, Z::ZeroSet{N})::Bool where {N<:Real}
 
 Check whether a given point is contained in a zero set.
 
@@ -75,7 +120,7 @@ julia> ∈([0.0, 0.0], Z)
 true
 ```
 """
-function ∈(x::AbstractVector{N}, Z::ZeroSet)::Bool where {N<:Real}
+function ∈(x::AbstractVector{N}, Z::ZeroSet{N})::Bool where {N<:Real}
     @assert length(x) == dim(Z)
 
     zero_N = zero(N)