diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 915bcba..7939bb0 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -27,26 +27,18 @@ jobs:
             os: ubuntu-latest
             arch: x64
     steps:
-      - uses: actions/checkout@v3
-      - uses: julia-actions/setup-julia@v1
+      - uses: actions/checkout@v4
+      - uses: julia-actions/setup-julia@v2
         with:
           version: ${{ matrix.version }}
           arch: ${{ matrix.arch }}
-      - uses: actions/cache@v1
-        env:
-          cache-name: cache-artifacts
-        with:
-          path: ~/.julia/artifacts
-          key: ${{ runner.os }}-test-${{ env.cache-name }}-${{ hashFiles('**/Project.toml') }}
-          restore-keys: |
-            ${{ runner.os }}-test-${{ env.cache-name }}-
-            ${{ runner.os }}-test-
-            ${{ runner.os }}-
+      - uses: julia-actions/cache@v2
       - uses: julia-actions/julia-buildpkg@v1
       - uses: julia-actions/julia-runtest@v1
         with:
           depwarn: error
       - uses: julia-actions/julia-processcoverage@v1
-      - uses: codecov/codecov-action@v3
+      - uses: codecov/codecov-action@v4
         with:
           file: lcov.info
+          token: ${{ secrets.CODECOV_TOKEN }}
diff --git a/Project.toml b/Project.toml
index 02c7d93..66db4d5 100644
--- a/Project.toml
+++ b/Project.toml
@@ -12,7 +12,7 @@ Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [compat]
-MultivariatePolynomials = "0.5.6"
+MultivariatePolynomials = "0.5.9"
 MutableArithmetics = "1"
 Reexport = "1"
 julia = "1"
diff --git a/src/comp.jl b/src/comp.jl
index a74c019..62ccfbb 100644
--- a/src/comp.jl
+++ b/src/comp.jl
@@ -2,7 +2,7 @@ import Base.==
 
 # TODO This should be in Base with T instead of Variable{V,M}.
 # See https://github.com/blegat/MultivariatePolynomials.jl/issues/3
-function (==)(x::Vector{Variable{V,M}}, y::Vector{Variable{V,M}}) where {V,M}
+function Base.:(==)(x::Vector{Variable{V,M}}, y::Vector{Variable{V,M}}) where {V,M}
     if length(x) != length(y)
         false
     else
@@ -20,119 +20,20 @@ end
 
 const AnyCommutative{O} = Union{Commutative{O},NonCommutative{O}}
 
-function (==)(
-    x::Variable{<:AnyCommutative{CreationOrder}},
-    y::Variable{<:AnyCommutative{CreationOrder}},
-)
-    return x.variable_order.order.id == y.variable_order.order.id &&
-           x.kind == y.kind
-end
-
-function Base.isless(
+function Base.cmp(
     x::Variable{<:AnyCommutative{CreationOrder}},
     y::Variable{<:AnyCommutative{CreationOrder}},
 )
     if x.variable_order.order.id == y.variable_order.order.id
-        return isless(y.kind, x.kind)
+        return cmp(y.kind, x.kind)
     else
-        return isless(y.variable_order.order.id, x.variable_order.order.id)
-    end
-end
-
-# Comparison of Monomial
-
-function MP.compare(x::Monomial{V,M}, y::Monomial{V,M}) where {V,M}
-    return MP.compare(x, y, M)
-end
-
-function MP.compare(
-    x::Monomial{V},
-    y::Monomial{V},
-    ::Type{MP.InverseLexOrder},
-) where {V}
-    i = MP.nvariables(x)
-    j = MP.nvariables(y)
-    @inbounds while i >= 1 && j >= 1
-        if x.vars[i] < y.vars[j]
-            if x.z[i] == 0
-                i -= 1
-            else
-                return 1
-            end
-        elseif x.vars[i] > y.vars[j]
-            if y.z[j] == 0
-                j -= 1
-            else
-                return -1
-            end
-        elseif x.z[i] != y.z[j]
-            return x.z[i] - y.z[j]
-        else
-            i -= 1
-            j -= 1
-        end
+        return cmp(y.variable_order.order.id, x.variable_order.order.id)
     end
-    return 0
 end
 
-function MP.compare(
-    x::Monomial{V},
-    y::Monomial{V},
-    ::Type{MP.LexOrder},
-) where {V}
-    i = j = 1
-    @inbounds while i <= nvariables(x) && j <= nvariables(y)
-        if x.vars[i] > y.vars[j]
-            if x.z[i] == 0
-                i += 1
-            else
-                return 1
-            end
-        elseif x.vars[i] < y.vars[j]
-            if y.z[j] == 0
-                j += 1
-            else
-                return -1
-            end
-        elseif x.z[i] != y.z[j]
-            return x.z[i] - y.z[j]
-        else
-            i += 1
-            j += 1
-        end
-    end
-    @inbounds while i <= nvariables(x)
-        if x.z[i] > 0
-            return 1
-        end
-	i += 1
-    end
-    @inbounds while j <= nvariables(y)
-        if y.z[j] > 0
-            return -1
-        end
-	j += 1
-    end
-    return 0
-end
-
-function (==)(x::Monomial{V,M}, y::Monomial{V,M}) where {V,M}
-    return MP.compare(x, y) == 0
-end
-function (==)(x::Variable{V,M}, y::Monomial{V,M}) where {V,M}
-    return convert(Monomial{V,M}, x) == y
-end
-
-# graded lex ordering
-function Base.isless(x::Monomial{V,M}, y::Monomial{V,M}) where {V,M}
-    return MP.compare(x, y) < 0
-end
-function Base.isless(x::Monomial{V,M}, y::Variable{V,M}) where {V,M}
-    return isless(x, convert(Monomial{V,M}, y))
-end
-function Base.isless(x::Variable{V,M}, y::Monomial{V,M}) where {V,M}
-    return isless(convert(Monomial{V,M}, x), y)
-end
+# TODO remove
+Base.:(==)(x::Variable, y::Variable) = iszero(cmp(x, y))
+Base.:(==)(x::Monomial, y::Monomial) = iszero(cmp(x, y))
 
 # Comparison of MonomialVector
 function (==)(x::MonomialVector{V,M}, y::MonomialVector{V,M}) where {V,M}
@@ -143,8 +44,8 @@ function (==)(x::MonomialVector{V,M}, y::MonomialVector{V,M}) where {V,M}
     # Should be sorted in the same order since the non-common
     # polyvar should have exponent 0
     for (a, b) in zip(x.Z, y.Z)
-        A = zeros(length(allvars))
-        B = zeros(length(allvars))
+        A = zeros(Int, length(allvars))
+        B = zeros(Int, length(allvars))
         A[maps[1]] = a
         B[maps[2]] = b
         if A != B
diff --git a/src/monomial_vector.jl b/src/monomial_vector.jl
index b9d3e51..4f3520b 100644
--- a/src/monomial_vector.jl
+++ b/src/monomial_vector.jl
@@ -148,62 +148,6 @@ function _error_for_negative_degree(deg)
     end
 end
 
-const _Lex = Union{MP.LexOrder,MP.InverseLexOrder}
-
-_last_lex_index(n, ::Type{MP.LexOrder}) = n
-_prev_lex_index(i, ::Type{MP.LexOrder}) = i - 1
-_not_first_indices(n, ::Type{MP.LexOrder}) = n:-1:2
-_last_lex_index(_, ::Type{MP.InverseLexOrder}) = 1
-_prev_lex_index(i, ::Type{MP.InverseLexOrder}) = i + 1
-_not_first_indices(n, ::Type{MP.InverseLexOrder}) = 1:(n-1)
-
-function _fill_exponents!(Z, n, degs, ::Type{Commutative}, M::Type{<:_Lex}, filter::Function)
-    _error_for_negative_degree.(degs)
-    maxdeg = maximum(degs, init = 0)
-    I = _not_first_indices(n, M)
-    z = zeros(Int, n)
-    while true
-        deg = sum(z)
-        if deg in degs && filter(z)
-            push!(Z, z)
-            z = copy(z)
-        end
-        if deg == maxdeg
-            i = findfirst(i -> !iszero(z[i]), I)
-            if isnothing(i)
-                break
-            end
-            j = I[i]
-            z[j] = 0
-            z[_prev_lex_index(j, M)] += 1
-        else
-            z[_last_lex_index(n, M)] += 1
-        end
-    end
-end
-
-function _fill_exponents!(Z, n, deg, ::Type{Commutative}, M::Type{<:_Lex}, filter::Function, ::Int)
-    _error_for_negative_degree(deg)
-    I = _not_first_indices(n, M)
-    z = zeros(Int, n)
-    z[_last_lex_index(n, M)] = deg
-    while true
-        if filter(z)
-            push!(Z, z)
-            z = copy(z)
-        end
-        i = findfirst(i -> !iszero(z[i]), I)
-        if isnothing(i)
-            break
-        end
-        j = I[i]
-        p = z[j]
-        z[j] = 0
-        z[_last_lex_index(n, M)] = p - 1
-        z[_prev_lex_index(j, M)] += 1
-    end
-end
-
 function _fill_noncomm_exponents_rec!(Z, z, i, n, deg, ::Type{MP.LexOrder}, filter::Function)
     if deg == 0
         if filter(z)
@@ -235,13 +179,6 @@ function _fill_exponents!(
     return reverse!(view(Z, start:length(Z)))
 end
 
-function _fill_exponents!(Z, n, deg, ::Type{V}, ::Type{MP.Reverse{M}}, args...) where {V,M}
-    prev = lastindex(Z)
-    _fill_exponents!(Z, n, deg, V, M, args...)
-    reverse!(view(Z, (prev + 1):lastindex(Z)))
-    return
-end
-
 function _fill_exponents!(
     Z::Vector{Vector{Int}},
     n,
@@ -266,7 +203,7 @@ function _all_exponents(
     ::Type{M},
     filter::Function,
 ) where {V,M}
-    Z = Vector{Vector{Int}}()
+    Z = Vector{Int}[]
     _fill_exponents!(Z, n, degs, V, M, filter)
     _isless = let M = M
         (a, b) -> MP.compare(a, b, M) < 0
@@ -275,28 +212,11 @@ function _all_exponents(
     return Z
 end
 
-function MonomialVector(
-    vars::Vector{<:Variable{<:Commutative,M}},
-    degs::AbstractVector{Int},
-    filter::Function = x -> true,
-) where {M}
-    vars = unique!(sort(vars, rev = true))
-    return MonomialVector(
-        vars,
-        _all_exponents(
-            length(vars),
-            degs,
-            Commutative,
-            M,
-            z -> filter(Monomial(vars, z)),
-        ),
-    )
-end
-
 function getvarsforlength(vars::Vector{<:Variable{<:NonCommutative}}, len::Int)
     n = length(vars)
     return map(i -> vars[((i-1)%n)+1], 1:len)
 end
+
 function MonomialVector(
     vars::Vector{<:Variable{<:NonCommutative,M}},
     degs::AbstractVector{Int},
@@ -313,6 +233,31 @@ function MonomialVector(
     v = isempty(Z) ? vars : getvarsforlength(vars, length(first(Z)))
     return MonomialVector(v, Z)
 end
+
+function MonomialVector(
+    vars::Vector{<:Variable{<:Commutative,M}},
+    degs::AbstractVector{Int},
+    filter::Function = x -> true,
+) where {M}
+    vars = unique!(sort(vars, rev = true))
+    if isempty(degs)
+        mindegree = 0
+        maxdegree = -1
+    else
+        mindegree = minimum(degs)
+        maxdegree = maximum(degs)
+    end
+    Z = Iterators.Filter(MP.ExponentsIterator{M}(
+        zeros(Int, length(vars));
+        mindegree,
+        maxdegree,
+    )) do z
+        mono = Monomial(vars, z)
+        MP.degree(mono) in degs && filter(mono)
+    end
+    return MonomialVector(vars, collect(Z))
+end
+
 function MonomialVector(
     vars::Vector{<:Variable},
     degs::Int,
@@ -324,6 +269,7 @@ end
 function MP.monomials(vars::AbstractVector{<:Variable}, args...)
     return MonomialVector(vars, args...)
 end
+
 function MP.monomials(vars::Tuple{Vararg{Variable}}, args...)
     return monomials([vars...], args...)
 end
@@ -390,7 +336,7 @@ end
 function MonomialVector{V,M}(X::DMonoVec{V,M}) where {V,M}
     allvars, Z = buildZvarsvec(Variable{V,M}, X)
     _isless = let M = M
-        (a, b) -> MP.compare(a, b, M) < 0
+        (a, b) -> cmp(M(), a, b) < 0
     end
     sort!(Z, lt = _isless)
     dups = findall(i -> Z[i] == Z[i-1], 2:length(Z))
diff --git a/src/operators.jl b/src/operators.jl
index 1f999c8..e5d4e0c 100644
--- a/src/operators.jl
+++ b/src/operators.jl
@@ -109,7 +109,7 @@ function MA.operate!(
 end
 
 function _exponents_compare(q::Polynomial{V,M}, j, e) where {V,M}
-    return MP.compare(q.x.Z[j], e, M)
+    return cmp(M(), q.x.Z[j], e)
 end
 
 # TODO need to check that this also works for non-commutative
diff --git a/src/poly.jl b/src/poly.jl
index 8334416..119d608 100644
--- a/src/poly.jl
+++ b/src/poly.jl
@@ -219,7 +219,7 @@ function removedups_to!(
     ::Type{M},
 ) where {T,M}
     _isless = let M = M
-        (a, b) -> MP.compare(a, b, M) < 0
+        (a, b) -> cmp(M(), a, b) < 0
     end
     σ = sortperm(Zdup, lt = _isless)
     i = 0
diff --git a/src/promote.jl b/src/promote.jl
index 05eb398..ab07a94 100644
--- a/src/promote.jl
+++ b/src/promote.jl
@@ -1,3 +1,15 @@
+function MP.promote_variables(m1::Monomial, m2::Monomial)
+    if MP.variables(m1) == MP.variables(m2)
+        return m1, m2
+    end
+    allvars, maps = mergevars([MP.variables(m1), MP.variables(m2)])
+    z1 = zeros(Int, length(allvars))
+    z1[maps[1]] = m1.z
+    z2 = zeros(Int, length(allvars))
+    z2[maps[2]] = m2.z
+    return Monomial(allvars, z1), Monomial(allvars, z2)
+end
+
 function MP.promote_rule_constant(
     ::Type{T},
     ::Type{<:DMonomialLike{V,M}},
diff --git a/test/comp.jl b/test/comp.jl
index ff425c8..94c526b 100644
--- a/test/comp.jl
+++ b/test/comp.jl
@@ -13,8 +13,8 @@ end
 function _less(a, b)
     @test a < b
     @test b > a
-    @test compare(monomial(a), b) < 0
-    @test compare(b, monomial(a)) > 0
+    @test cmp(monomial(a), b) < 0
+    @test cmp(b, monomial(a)) > 0
 end
 @testset "Issue 152" begin
     @polyvar x y monomial_order=LexOrder
@@ -76,24 +76,15 @@ end
     _test_monomials([x, y], 3, [x^3, x^2 * y, x * y^2, y^3])
 end
 
-@testset "Reverse{LexOrder}" begin
-    @polyvar x y monomial_order = Reverse{LexOrder}
-    _test_less(y^2, y)
-    _test_less(y, x^0)
+@testset "Graded{Reverse{LexOrder}}" begin
+    @polyvar x y monomial_order = Graded{Reverse{LexOrder}}
+    _test_less(y, y^2)
+    _test_less(x^0, y)
     _test_less(x, y^2)
     _test_less(x^2, x * y^2)
     _test_monomials([x, y], 3, [x^3, x^2 * y, x * y^2, y^3])
-end
-
-@testset "Reverse{InverseLexOrder}" begin
-    @polyvar x y monomial_order = Reverse{InverseLexOrder}
-    _test_less(y^2, y)
-    _test_less(y, x^0)
-    _test_less(y^2, x)
-    _test_less(x * y^2, x^2)
-    _test_monomials([x, y], 3, [y^3, y^2 * x, y * x^2, x^3])
-    _test_monomials([x, y], 1:2, [y^2, x * y, y, x^2, x])
-    _test_monomials([x, y], [0, 1, 3], [y^3, x*y^2, x^2*y, y, x^3, x, 1])
+    _test_monomials([x, y], 1:2, [x, y, x^2, x * y, y^2])
+    _test_monomials([x, y], [0, 1, 3], [1, x, y, x^3, x^2 * y, x * y^2, y^3])
 end
 
 @testset "Graded{Reverse{InverseLexOrder}}" begin