diff --git a/src/comparison.jl b/src/comparison.jl
index cfa9cb9..430533a 100644
--- a/src/comparison.jl
+++ b/src/comparison.jl
@@ -311,18 +311,13 @@ Base.@pure function Base.cmp(v1::AbstractVariable, v2::AbstractVariable)
     return -cmp(name(v1), name(v2))
 end
 
-function Base.cmp(m1::AbstractMonomial, m2::AbstractMonomial)
-    s1, s2 = promote_variables(m1, m2)
-    return cmp(ordering(m1)(), exponents(s1), exponents(s2))
+function Base.:(==)(m1::AbstractMonomialLike, m2::AbstractMonomialLike)
+    return iszero(cmp(m1, m2))
 end
 
-function compare(
-    m1::AbstractMonomial,
-    m2::AbstractMonomial,
-    ::Type{O},
-) where {O<:AbstractMonomialOrdering}
+function Base.cmp(m1::AbstractMonomial, m2::AbstractMonomial)
     s1, s2 = promote_variables(m1, m2)
-    return cmp(O(), exponents(s1), exponents(s2))
+    return cmp(ordering(m1)(), exponents(s1), exponents(s2))
 end
 
 # Implement this to make coefficients be compared with terms.
@@ -344,7 +339,7 @@ function Base.cmp(t1::AbstractTermLike, t2::AbstractTermLike)
     return Δ
 end
 
-Base.isless(t1::AbstractTermLike, t2::AbstractTermLike) = compare(t1, t2) < 0
+Base.isless(t1::AbstractTermLike, t2::AbstractTermLike) = cmp(t1, t2) < 0
 
 """
     struct ExponentsIterator{M}(
diff --git a/src/default_polynomial.jl b/src/default_polynomial.jl
index a7b45bf..a02c113 100644
--- a/src/default_polynomial.jl
+++ b/src/default_polynomial.jl
@@ -93,7 +93,7 @@ Base.one(p::Polynomial) = one(typeof(p))
 Base.zero(::Type{Polynomial{C,T,A}}) where {C,T,A} = Polynomial{C,T,A}(A())
 Base.zero(t::Polynomial) = zero(typeof(t))
 
-compare_monomials(a, b) = compare(monomial(a), monomial(b))
+compare_monomials(a, b) = cmp(monomial(a), monomial(b))
 
 function join_terms(
     terms1::AbstractArray{<:Term},
@@ -294,7 +294,7 @@ function __polynomial_merge!(
             if tp isa Int && j isa Int
                 tp = get1(tp)
             end
-            compare(monomial(*(monomials(q)[j], monomial.(t)...)), monomial(tp))
+            cmp(monomial(*(monomials(q)[j], monomial.(t)...)), monomial(tp))
         end
     end
     get2 = let t = t, q = q
@@ -385,7 +385,7 @@ function __polynomial_merge!(
             if tp isa Int && j isa Int
                 tp = get1(tp)
             end
-            compare(monomial(monomial(t) * monomials(q)[j]), monomial(tp))
+            cmp(monomial(monomial(t) * monomials(q)[j]), monomial(tp))
         end
     end
     get2 = let t = t, q = q
@@ -460,7 +460,7 @@ function __polynomial_merge!(
             if t isa Int && j isa Int
                 t = get1(t)
             end
-            compare(monomials(q)[j], monomial(t))
+            cmp(monomials(q)[j], monomial(t))
         end
     end
     get2 = let q = q
diff --git a/src/sequences.jl b/src/sequences.jl
index deee5c1..6106e65 100644
--- a/src/sequences.jl
+++ b/src/sequences.jl
@@ -18,7 +18,7 @@ function merge_sorted!(
     while i1 <= lastindex(v1) && i2 <= lastindex(v2)
         x1 = v1[i1]
         x2 = v2[i2]
-        Δ = compare(x1, x2)
+        Δ = cmp(x1, x2)
         if Δ < 0
             if filter(x1)
                 result[i] = x1