diff --git a/shardy/dialect/sdy/ir/attrs.td b/shardy/dialect/sdy/ir/attrs.td
index ce29e4e3..a3226a8b 100644
--- a/shardy/dialect/sdy/ir/attrs.td
+++ b/shardy/dialect/sdy/ir/attrs.td
@@ -181,12 +181,26 @@ def Sdy_AxisRef : AttrDef<Sdy_Dialect, "AxisRef"> {
     // For example:
     //  "a":(1)2, "a"      -> true
     //  "a":(2)2, "a":(2)4 -> true
+    //  "a", "a"           -> true
     //  "a":(2)4, "a":(2)4 -> true
     //  "a":(1)4, "a":(1)2 -> false
     //  "a":(1)4, "a":(2)8 -> false
     //  "a":(1)2, "b"      -> false
     bool prefixOf(AxisRefAttr other) const;
 
+    // Returns whether this axis or sub-axis is a strict prefix of `other`.
+    // "a.strictPrefixOf(b)" is equivalent to "a.prefixOf(b) && a != b".
+    //
+    // For example:
+    //  "a":(1)2, "a"      -> true
+    //  "a":(2)2, "a":(2)4 -> true
+    //  "a", "a"           -> false
+    //  "a":(2)4, "a":(2)4 -> false
+    //  "a":(1)4, "a":(1)2 -> false
+    //  "a":(1)4, "a":(2)8 -> false
+    //  "a":(1)2, "b"      -> false
+    bool strictPrefixOf(AxisRefAttr other) const;
+
     // Returns whether this axis or sub-axis overlaps with `other`, i.e., they
     // are equal or there is a sub-axis that is contained in both axis refs.
     //
diff --git a/shardy/dialect/sdy/ir/dialect.cc b/shardy/dialect/sdy/ir/dialect.cc
index 64482df6..f6f88bc5 100644
--- a/shardy/dialect/sdy/ir/dialect.cc
+++ b/shardy/dialect/sdy/ir/dialect.cc
@@ -183,6 +183,10 @@ bool AxisRefAttr::prefixOf(AxisRefAttr other) const {
          getSubAxisPreSize() == other.getSubAxisPreSize();
 }
 
+bool AxisRefAttr::strictPrefixOf(AxisRefAttr other) const {
+  return prefixOf(other) && *this != other;
+}
+
 bool AxisRefAttr::overlaps(AxisRefAttr other) const {
   if (other.getName() != getName()) {
     return false;
diff --git a/shardy/dialect/sdy/ir/dialect_test.cc b/shardy/dialect/sdy/ir/dialect_test.cc
index 5cecdf40..6c5c595e 100644
--- a/shardy/dialect/sdy/ir/dialect_test.cc
+++ b/shardy/dialect/sdy/ir/dialect_test.cc
@@ -102,18 +102,36 @@ TEST_F(DialectTest, AxisRefAttrContains) {
 }
 
 TEST_F(DialectTest, AxisRefAttrPrefixOf) {
-  EXPECT_TRUE(createAxis("x").prefixOf(createAxis("x")));
-  EXPECT_TRUE(createSubAxis("x", 1, 4).prefixOf(createAxis("x")));
-  EXPECT_TRUE(createSubAxis("x", 1, 2).prefixOf(createSubAxis("x", 1, 4)));
-  EXPECT_TRUE(createSubAxis("x", 2, 4).prefixOf(createSubAxis("x", 2, 4)));
-  EXPECT_TRUE(createSubAxis("x", 2, 2).prefixOf(createSubAxis("x", 2, 8)));
-
-  EXPECT_FALSE(createAxis("x").prefixOf(createAxis("y")));
-  EXPECT_FALSE(createSubAxis("x", 1, 2).prefixOf(createAxis("y")));
-  EXPECT_FALSE(createAxis("x").prefixOf(createSubAxis("x", 1, 2)));
-  EXPECT_FALSE(createSubAxis("x", 1, 4).prefixOf(createSubAxis("x", 1, 2)));
-  EXPECT_FALSE(createSubAxis("x", 1, 4).prefixOf(createSubAxis("x", 4, 2)));
-  EXPECT_FALSE(createSubAxis("x", 1, 4).prefixOf(createSubAxis("x", 2, 4)));
+  auto strictPrefixOf = [](AxisRefAttr a, AxisRefAttr b) {
+    EXPECT_TRUE(a.strictPrefixOf(b));
+    EXPECT_TRUE(a.prefixOf(b));
+    EXPECT_FALSE(b.prefixOf(a));
+    EXPECT_FALSE(b.strictPrefixOf(a));
+  };
+  strictPrefixOf(createSubAxis("x", 1, 4), createAxis("x"));
+  strictPrefixOf(createSubAxis("x", 1, 2), createSubAxis("x", 1, 4));
+  strictPrefixOf(createSubAxis("x", 2, 2), createSubAxis("x", 2, 8));
+
+  auto equals = [](AxisRefAttr a, AxisRefAttr b) {
+    EXPECT_TRUE(a == b);
+    EXPECT_TRUE(a.prefixOf(b));
+    EXPECT_TRUE(b.prefixOf(a));
+    EXPECT_FALSE(a.strictPrefixOf(b));
+    EXPECT_FALSE(b.strictPrefixOf(a));
+  };
+  equals(createAxis("x"), createAxis("x"));
+  equals(createSubAxis("x", 2, 4), createSubAxis("x", 2, 4));
+
+  auto isNotPrefix = [](AxisRefAttr a, AxisRefAttr b) {
+    EXPECT_FALSE(a.prefixOf(b));
+    EXPECT_FALSE(b.prefixOf(a));
+    EXPECT_FALSE(a.strictPrefixOf(b));
+    EXPECT_FALSE(b.strictPrefixOf(a));
+  };
+  isNotPrefix(createAxis("x"), createAxis("y"));
+  isNotPrefix(createSubAxis("x", 1, 2), createAxis("y"));
+  isNotPrefix(createSubAxis("x", 1, 4), createSubAxis("x", 4, 2));
+  isNotPrefix(createSubAxis("x", 1, 4), createSubAxis("x", 2, 4));
 }
 
 TEST_F(DialectTest, AxisRefAttrOverlaps) {