diff --git a/mlir/include/mlir/Analysis/Presburger/PresburgerRelation.h b/mlir/include/mlir/Analysis/Presburger/PresburgerRelation.h
--- a/mlir/include/mlir/Analysis/Presburger/PresburgerRelation.h
+++ b/mlir/include/mlir/Analysis/Presburger/PresburgerRelation.h
@@ -131,6 +131,17 @@
   /// false otherwise.
   bool isIntegerEmpty() const;
 
+  /// Return true if there is no disjunct, false otherwise.
+  bool isPlainEmpty() const;
+
+  /// Return true if the set is known to have one unconstrained disjunct, false
+  /// otherwise.
+  bool isPlainUniverse() const;
+
+  /// Return true if the set is consist of a single disjunct, without any local
+  /// variables, false otherwise.
+  bool isConvexNoLocals() const;
+
   /// Find an integer sample from the given set. This should not be called if
   /// any of the disjuncts in the union are unbounded.
   bool findIntegerSample(SmallVectorImpl<MPInt> &sample);
diff --git a/mlir/lib/Analysis/Presburger/PresburgerRelation.cpp b/mlir/lib/Analysis/Presburger/PresburgerRelation.cpp
--- a/mlir/lib/Analysis/Presburger/PresburgerRelation.cpp
+++ b/mlir/lib/Analysis/Presburger/PresburgerRelation.cpp
@@ -98,6 +98,14 @@
 PresburgerRelation::intersect(const PresburgerRelation &set) const {
   assert(space.isCompatible(set.getSpace()) && "Spaces should match");
 
+  // If the set is empty or the other set is universe,
+  // directly return the set
+  if (isPlainEmpty() || set.isPlainUniverse())
+    return *this;
+
+  if (set.isPlainEmpty() || isPlainUniverse())
+    return set;
+
   PresburgerRelation result(getSpace());
   for (const IntegerRelation &csA : disjuncts) {
     for (const IntegerRelation &csB : set.disjuncts) {
@@ -495,6 +503,27 @@
   return this->isSubsetOf(set) && set.isSubsetOf(*this);
 }
 
+/// Return true if the Presburger relation represents the universe set, false
+/// otherwise. It is a simple check that only check if the relation has at least
+/// one unconstrained disjunct, indicating the absence of constraints or
+/// conditions.
+bool PresburgerRelation::isPlainUniverse() const {
+  for (auto &disjunct : getAllDisjuncts()) {
+    if (disjunct.getNumConstraints() == 0)
+      return true;
+  }
+  return false;
+}
+
+bool PresburgerRelation::isConvexNoLocals() const {
+  if (getNumDisjuncts() == 1 && getSpace().getNumLocalVars() == 0)
+    return true;
+  return false;
+}
+
+/// Return true if there is no disjunct, false otherwise.
+bool PresburgerRelation::isPlainEmpty() const { return getNumDisjuncts() == 0; }
+
 /// Return true if all the sets in the union are known to be integer empty,
 /// false otherwise.
 bool PresburgerRelation::isIntegerEmpty() const {