diff --git a/src/core/PocomathInstance.mjs b/src/core/PocomathInstance.mjs
index f8d9c5a..398bcee 100644
--- a/src/core/PocomathInstance.mjs
+++ b/src/core/PocomathInstance.mjs
@@ -30,9 +30,12 @@ export default class PocomathInstance {
       'install',
       'installType',
       'instantiateTemplate',
+      'isSubtypeOf',
       'joinTypes',
       'name',
+      'returnTypeOf',
       'self',
+      'subtypesOf',
       'Templates',
       'typeOf',
       'Types',
@@ -56,7 +59,8 @@ export default class PocomathInstance {
       this.Templates = {}
       // The actual type testing functions
       this._typeTests = {}
-      this._subtypes = {} // For each type, gives all of its (in)direct subtypes
+      // For each type, gives all of its (in)direct subtypes in topo order:
+      this._subtypes = {}
       /* The following gives for each type, a set of all types that could
        * match in typed-function's dispatch algorithm before the given type.
        * This is important because if we instantiate a template, we must
@@ -334,14 +338,14 @@ export default class PocomathInstance {
       this._typeTests[type] = testFn
       this._typed.addTypes([{name: type, test: testFn}], beforeType)
       this.Types[type] = spec
-      this._subtypes[type] = new Set()
+      this._subtypes[type] = []
       this._priorTypes[type] = new Set()
       // Update all the subtype sets of supertypes up the chain
       let nextSuper = spec.refines
       while (nextSuper) {
          this._invalidateDependents(':' + nextSuper)
          this._priorTypes[nextSuper].add(type)
-         this._subtypes[nextSuper].add(type)
+         this._addSubtypeTo(nextSuper, type)
          nextSuper = this.Types[nextSuper].refines
       }
       /* Now add conversions to this type */
@@ -369,8 +373,8 @@ export default class PocomathInstance {
          for (const fromtype in this.Types[to].from) {
             if (type == fromtype
                 || (fromtype in this._subtypes
-                    && this._subtypes[fromtype].has(type))) {
-               if (spec.refines == to || spec.refines in this._subtypes[to]) {
+                    && this.isSubtypeOf(type, fromtype))) {
+               if (spec.refines == to || this.isSubtypeOf(spec.refines,to)) {
                   throw new SyntaxError(
                      `Conversion of ${type} to its supertype ${to} disallowed.`)
                }
@@ -398,8 +402,30 @@ export default class PocomathInstance {
       this._installFunctions({typeOf: imp})
    }
 
-   /* Returns the most refined type of all the types in the array, with
-    * '' standing for the empty type for convenience. If the second
+   _addSubtypeTo(sup, sub) {
+      if (this.isSubtypeOf(sub, sup)) return
+      const supSubs = this._subtypes[sup]
+      let i
+      for (i = 0; i < supSubs.length; ++i) {
+         if (this.isSubtypeOf(sub, supSubs[i])) break
+      }
+      supSubs.splice(i, 0, sub)
+   }
+
+   /* Returns true is typeA is a subtype of type B */
+   isSubtypeOf(typeA, typeB) {
+      return this._subtypes[typeB].includes(typeA)
+   }
+
+   /* Returns a list of the subtypes of a given type, in topological sorted
+    * order (i.e, no type on the list contains one that comes after it).
+    */
+   subtypesOf(type) {
+      // HERE! For this to work, have to maintain subtypes as a sorted list.
+      return this._subtypes[type] // should we clone?
+   }
+   /* Returns the most refined type containing all the types in the array,
+    * with '' standing for the empty type for convenience. If the second
     * argument `convert` is true, a convertible type is considered a
     * a subtype (defaults to false).
     */
@@ -418,19 +444,20 @@ export default class PocomathInstance {
       if (typeA === 'ground' || typeB === 'ground') return 'ground'
       if (typeA === typeB) return typeA
       const subber = convert ? this._priorTypes : this._subtypes
-      if (subber[typeB].has(typeA)) return typeB
+      const pick = convert ? 'has' : 'includes'
+      if (subber[typeB][pick](typeA)) return typeB
       /* OK, so we need the most refined supertype of A that contains B:
        */
       let nextSuper = typeA
       while (nextSuper) {
-         if (subber[nextSuper].has(typeB)) return nextSuper
+         if (subber[nextSuper][pick](typeB)) return nextSuper
          nextSuper = this.Types[nextSuper].refines
       }
       /* And if conversions are allowed, we have to search the other way too */
       if (convert) {
          nextSuper = typeB
          while (nextSuper) {
-            if (subber[nextSuper].has(typeA)) return nextSuper
+            if (subber[nextSuper][pick](typeA)) return nextSuper
             nextSuper = this.Types[nextSuper].refines
          }
       }
@@ -1132,7 +1159,7 @@ export default class PocomathInstance {
                break
             }
             if (myType === otherType
-                || this._subtypes[otherType].has(myType)) {
+                || this.isSubtypeOf(myType, otherType)) {
                continue
             }
             if (otherType in this.Templates) {
diff --git a/test/_pocomath.mjs b/test/_pocomath.mjs
index 9e52a25..c5e8969 100644
--- a/test/_pocomath.mjs
+++ b/test/_pocomath.mjs
@@ -22,6 +22,7 @@ describe('The default full pocomath instance "math"', () => {
 
    it('can determine the return types of operations', () => {
       assert.strictEqual(math.returnTypeOf('negate', 'number'), 'number')
+      assert.strictEqual(math.returnTypeOf('negate', 'NumInt'), 'NumInt')
    })
 
    it('can subtract numbers', () => {