diff --git a/Project.toml b/Project.toml
index d2db3d5a..62a301d5 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "InitialValues"
 uuid = "22cec73e-a1b8-11e9-2c92-598750a2cf9c"
 authors = ["Takafumi Arakaki <aka.tkf@gmail.com>"]
-version = "0.1.0"
+version = "0.2.0-DEV"
 
 [compat]
 julia = "1.0"
diff --git a/src/InitialValues.jl b/src/InitialValues.jl
index ae8168a3..311a4d69 100644
--- a/src/InitialValues.jl
+++ b/src/InitialValues.jl
@@ -123,6 +123,13 @@ def_impl(op, x, y) =
         InitialValues.hasinitialvalue(::Type{typeof($op)}) = true
     end
 
+monoid_impl(op, x) =
+    quote
+        $op(::$(itypeof_impl(op)), $x::$(itypeof_impl(op))) = $x
+        $op($x, ::$(itypeof_impl(op))) = $x
+        $(def_impl(op, x, x))
+    end
+
 """
     InitialValues.@def op [y = :x]
 
@@ -149,34 +156,49 @@ macro def(op, y = :x)
     def_impl(esc(op), esc(:x), esc(y))
 end
 
-disambiguate_impl(op, right, x, y) =
+"""
+    InitialValues.@monoid op
+
+Define a generic identity for a binary operator `op`.
+`InitialValues.@monoid op` is expanded to
+
+```julia
+$(prettyexpr(monoid_impl(:op, :x)))
+```
+"""
+macro monoid(op)
+    monoid_impl(esc(op), esc(:x))
+end
+
+disambiguate_impl(op, other, x) =
     quote
-        $op(::$(itypeof_impl(op)), $x::$right) = $y
+        $op(::$(itypeof_impl(op)), $x::$other) = $x
+        $op($x::$other, ::$(itypeof_impl(op))) = $x
     end
 
 """
-    InitialValues.@disambiguate op RightType [y = :x]
+    InitialValues.@disambiguate op OtherType
 
-Disambiguate the method introduced by [`@def`](@ref).
+Disambiguate the method introduced by [`@monoid`](@ref).
 
 It is expanded to
 
 ```julia
-$(prettyexpr(disambiguate_impl(:op, :RightType, :x, :x)))
+$(prettyexpr(disambiguate_impl(:op, :OtherType, :x)))
 ```
 """
-macro disambiguate(op, right, y = :x)
-    disambiguate_impl(esc(op), esc(right), esc(:x), esc(y))
+macro disambiguate(op, other)
+    disambiguate_impl(esc(op), esc(other), esc(:x))
 end
 
-@def Base.:*
-@def Base.:+
-@def Base.:&
-@def Base.:|
-@def Base.min
-@def Base.max
-@def Base.add_sum
-@def Base.mul_prod
+@monoid Base.:*
+@monoid Base.:+
+@monoid Base.:&
+@monoid Base.:|
+@monoid Base.min
+@monoid Base.max
+@monoid Base.add_sum
+@monoid Base.mul_prod
 
 @disambiguate Base.min Missing
 @disambiguate Base.max Missing
diff --git a/test/test_def.jl b/test/test_def.jl
index 92d067e3..3595cf9d 100644
--- a/test/test_def.jl
+++ b/test/test_def.jl
@@ -8,7 +8,7 @@ module CleanNameSpace
     add(x, y) = x + y
     got(x) = string("Got: ", repr(x))
     @def add got(x)
-    @disambiguate add Missing got(x)
+    @disambiguate add Missing
 end
 
 @testset "CleanNameSpace" begin
@@ -17,7 +17,7 @@ end
     @test hasinitialvalue(add)
     @test isknown(Init(add))
     @test add(Init(add), :x) == "Got: :x"
-    @test add(Init(add), missing) == "Got: missing"
+    @test add(Init(add), missing) === missing
 end
 
 module NonFunction
@@ -26,7 +26,7 @@ module NonFunction
     const add = Add()
     add(x, y) = x + y
     @def add
-    @disambiguate add Missing "Got: $x"
+    @disambiguate add Missing
 end
 
 @testset "NonFunction" begin
@@ -35,7 +35,7 @@ end
     @test hasinitialvalue(add)
     @test isknown(Init(add))
     @test add(Init(add), :x) == :x
-    @test add(Init(add), missing) == "Got: missing"
+    @test add(Init(add), missing) === missing
 end
 
 end  # module
diff --git a/test/test_prettyexpr.jl b/test/test_prettyexpr.jl
index 1da81dbf..c1493028 100644
--- a/test/test_prettyexpr.jl
+++ b/test/test_prettyexpr.jl
@@ -32,7 +32,7 @@ eq(x, y) = normalize(x) == normalize(y)
     ),
     (
         label = "disambiguate_impl",
-        ex = disambiguate_impl(:op, :RightType, :x, :x),
+        ex = disambiguate_impl(:op, :RightType, :x),
     ),
 ]
     code = prettyexpr(ex)