diff --git a/src/boolean/BooleanT.js b/src/boolean/BooleanT.js
index 8d26974..f2792ed 100644
--- a/src/boolean/BooleanT.js
+++ b/src/boolean/BooleanT.js
@@ -1,3 +1,6 @@
 import {Type} from '#core/Type.js'
 
-export const BooleanT = new Type(n => typeof n === 'boolean')
+export const BooleanT = new Type(n => typeof n === 'boolean', {
+   zero: false,
+   one: true
+})
diff --git a/src/core/README.md b/src/core/README.md
index 10d3401..0cdfd69 100644
--- a/src/core/README.md
+++ b/src/core/README.md
@@ -13,6 +13,13 @@ As of this writing, the only two types required to be in a TypeDispatcher are
 Undefined (the type inhabited only by `undefined`) and TypeOfTypes (the type
 inhabited exactly by Type objects).
 
+There is also a constant NotAType which is the type-world analogue of NaN for
+numbers. It is occasionally used for the rare behavior that truly does not
+return any particular type, such as the method `zero` that takes a Type and
+returns its zero element. However, it does not really work as a Type, and in
+particular, do _not_ merge it into any TypeDispatcher -- it will disrupt the
+type and method resolution process.
+
 ## Core methods
 
 Similarly, as of this writing the only methods that must be in a TypeDispatcher
diff --git a/src/core/Type.js b/src/core/Type.js
index 7c728a2..d95e5b7 100644
--- a/src/core/Type.js
+++ b/src/core/Type.js
@@ -2,14 +2,20 @@ export class Type {
    constructor(f, options = {}) {
       this.test = f
       this.from = options.from ?? {patterns: []} // mock empty Implementations
+      if ('zero' in options) this.zero = options.zero
+      if ('one' in options) this.one = options.one
    }
    toString() {
       return this.name || `[Type ${this.test}]`
    }
 }
 
-export const Undefined = new Type(t => typeof t === 'undefined')
+export const Undefined = new Type(
+   t => typeof t === 'undefined',
+   {zero: undefined, one: undefined})
 export const TypeOfTypes = new Type(t => t instanceof Type)
+export const NotAType = new Type(t => true) // Danger, do not merge!
+NotAType._doNotMerge = true
 
 export const Returns = (type, f) => (f.returns = type, f)
 
diff --git a/src/core/TypeDispatcher.js b/src/core/TypeDispatcher.js
index 503b577..fdcad33 100644
--- a/src/core/TypeDispatcher.js
+++ b/src/core/TypeDispatcher.js
@@ -1,7 +1,7 @@
 import ArrayKeyedMap from 'array-keyed-map'
 
 import {
-   Implementations, ImplementationsGenerator,
+   Implementations, ImplementationsGenerator, ResolutionError,
    isPlainFunction, isPlainObject, onType, types
 } from './helpers.js'
 import {bootstrapTypes, Returns, whichType, Type} from './Type.js'
@@ -80,6 +80,9 @@ export class TypeDispatcher {
 
          // Now dispatch on what sort of thing we are supposed to merge:
          if (val instanceof Type) {
+            if (val._doNotMerge) {
+               throw new TypeError(`attempt to merge unusable type '${val}'`)
+            }
             this.types[key] = val
             val.name = key
             continue
@@ -219,22 +222,21 @@ export class TypeDispatcher {
    // transforms them per the given template, to massage them into the form
    // expected by a behavior associated with the TypePattern that produced
    // the template.
-   _generateCollectFunction(template, state = {pos: 0}) {
-      const extractors = []
-      for (const elt of template) {
-         if (Array.isArray(elt)) {
-            extractors.push(this._generateCollectFunction(elt, state))
-         } else {
-            const from = state.pos++
-            if ('actual' in elt) { // incorporate conversion
-               let convert = elt.convertor
-               if (!convert.returns) { // it's a factory that produces convert
-                  convert = convert(this, elt.actual)
-               }
-               extractors.push(args => convert(args[from]))
-            } else extractors.push(args => args[from])
+   _generateCollectFunction(template, state=false) {
+      if (!Array.isArray(template)) {
+         const from = state ? state.pos++ : 0
+         let extractor = args => args[from]
+         if ('actual' in template) { // incorporate conversion
+            let convert = template.convertor
+            // Check if it's a factory:
+            if (!convert.returns) convert = convert(this, template.actual)
+            extractor = args => convert(args[from])
          }
+         return state ? extractor : args => [extractor(args)]
       }
+      state ||= {pos: 0}
+      const extractors = template.map(
+         item => this._generateCollectFunction(item, state))
       return args => extractors.map(f => f(args))
    }
 
@@ -284,7 +286,8 @@ export class TypeDispatcher {
          template = types
       }
       if (needItem) {
-         throw new TypeError(`no matching definition of '${key}' on '${types}'`)
+         throw new ResolutionError(
+            `no matching definition of '${key}' on '${types}'`)
       }
       // If this key is producing a non-function value, we're done
       if (!isPlainFunction(item)) {
@@ -316,7 +319,7 @@ export class TypeDispatcher {
                matched(template))
          } catch (e) {
             e.message = `Error in factory for ${key} on ${types} `
-               + `(match data ${template}): ${e}`
+               + `(match data ${template}): ${e.message}`
             throw e
          }
       } else theBehavior = item
diff --git a/src/core/__test__/TypeDispatcher.spec.js b/src/core/__test__/TypeDispatcher.spec.js
index 81d91a3..25ed721 100644
--- a/src/core/__test__/TypeDispatcher.spec.js
+++ b/src/core/__test__/TypeDispatcher.spec.js
@@ -3,9 +3,9 @@ import {TypeDispatcher} from '../TypeDispatcher.js'
 import * as booleans from '#boolean/all.js'
 import * as generics from '#generic/all.js'
 import * as numbers from '#number/all.js'
-import {onType} from "#core/helpers.js"
+import {onType, ResolutionError} from "#core/helpers.js"
 import {Any} from "#core/TypePatterns.js"
-import {Returns} from "#core/Type.js"
+import {Returns, NotAType} from "#core/Type.js"
 import {plain} from "#number/helpers.js"
 
 describe('TypeDispatcher', () => {
@@ -17,6 +17,7 @@ describe('TypeDispatcher', () => {
       const {NumberT, TypeOfTypes, Undefined} = incremental.types
       assert(NumberT.test(7))
       assert.strictEqual(incremental.add(-1.5, 0.5), -1)
+      assert.throws(() => incremental.add(7, undefined), ResolutionError)
       // Make Undefined act like zero:
       incremental.merge({add: onType(
          [Undefined, Any], (_m, [_U, T]) => Returns(T, (_a, b) => b),
@@ -30,7 +31,7 @@ describe('TypeDispatcher', () => {
       assert.strictEqual(
          incremental.add.resolve([Undefined, NumberT]).returns,
          NumberT)
-      // Oops, changed my mind, make it work like NaN with numbers:
+      // Oops, changed my mind ;-), make it work like NaN with numbers:
       const alwaysNaN = Returns(NumberT, () => NaN)
       incremental.merge({add: onType(
          [Undefined, NumberT], alwaysNaN,
@@ -63,4 +64,8 @@ describe('TypeDispatcher', () => {
       assert(!bgn._behaviors.negate.has([BooleanT]))
       assert.strictEqual(bgn.negate(true), -2)
    })
+   it('disallows merging NotAType', () => {
+      const doomed = new TypeDispatcher()
+      assert.throws(() => doomed.merge({NaT: NotAType}), TypeError)
+   })
 })
diff --git a/src/core/helpers.js b/src/core/helpers.js
index 8ecded5..f741859 100644
--- a/src/core/helpers.js
+++ b/src/core/helpers.js
@@ -36,6 +36,13 @@ export class ImplementationsGenerator {
 
 export const types = new ImplementationsGenerator(() => onType(Passthru, {}))
 
+export class ResolutionError extends TypeError {
+   constructor(...args) {
+      super(...args)
+      this.name = 'ResolutionError'
+   }
+}
+
 export const isPlainObject = obj => {
    if (typeof obj !== 'object') return false
    if (!obj) return false // excludes null
diff --git a/src/coretypes/__test__/utils.spec.js b/src/coretypes/__test__/utils.spec.js
new file mode 100644
index 0000000..732bf1f
--- /dev/null
+++ b/src/coretypes/__test__/utils.spec.js
@@ -0,0 +1,14 @@
+import assert from 'assert'
+import math from '#nanomath'
+
+describe('core type utility functions', () => {
+   it('identifies zero and one elements in most types', () => {
+      assert.strictEqual(math.zero(math.types.NumberT), 0)
+      assert.strictEqual(math.zero(math.types.Undefined), undefined)
+      assert.strictEqual(math.one(math.types.BooleanT), true)
+      assert.throws(() => math.one(math.types.TypeOfTypes), RangeError)
+      assert.strictEqual(math.one(-7.5), 1)
+      assert.strictEqual(math.one(undefined), undefined)
+      assert.strictEqual(math.zero(true), false)
+   })
+})
diff --git a/src/coretypes/all.js b/src/coretypes/all.js
index ff64f94..18b23af 100644
--- a/src/coretypes/all.js
+++ b/src/coretypes/all.js
@@ -1 +1,2 @@
 export * from './relational.js'
+export * from './utils.js'
diff --git a/src/coretypes/utils.js b/src/coretypes/utils.js
new file mode 100644
index 0000000..6bc35f0
--- /dev/null
+++ b/src/coretypes/utils.js
@@ -0,0 +1,26 @@
+import {onType} from '#core/helpers.js'
+import {NotAType, Returns, TypeOfTypes} from '#core/Type.js'
+import {Any} from "#core/TypePatterns.js"
+
+export const zero = onType(
+   Any, (math, T) => {
+      const z = math.zero(T)
+      return Returns(T, () => z)
+   },
+   TypeOfTypes, Returns(NotAType, t => {
+      if ('zero' in t) return t.zero
+      throw new RangeError(`type '${t}' has no zero element`)
+   })
+)
+
+export const one = onType(
+   Any, (math, T) => {
+      const unit = math.one(T)
+      return Returns(T, () => unit)
+   },
+   TypeOfTypes, Returns(NotAType, t => {
+      if ('one' in t) return t.one
+      throw new RangeError(
+         `type '${t}' has no unit element designated as "one"`)
+   })
+)
diff --git a/src/generic/__test__/relational.spec.js b/src/generic/__test__/relational.spec.js
index 5d61b76..6506f6a 100644
--- a/src/generic/__test__/relational.spec.js
+++ b/src/generic/__test__/relational.spec.js
@@ -2,6 +2,7 @@ import assert from 'assert'
 import math from '#nanomath'
 import * as numbers from '#number/all.js'
 import * as generics from '#generic/all.js'
+import {ResolutionError} from '#core/helpers.js'
 import {TypeDispatcher} from '#core/TypeDispatcher.js'
 
 describe('generic relational functions', () => {
@@ -38,4 +39,15 @@ describe('generic relational functions', () => {
       assert.strictEqual(equal(0, 1.1e-15), 1)
       assert.strictEqual(equal(0, 1.1e-13), 0)
    })
+   it('performs three-way comparison', () => {
+      const {compare} = math
+      assert.strictEqual(compare(-0.4e-15, +0.4e-15), 0)
+      assert.strictEqual(compare(2.2, true), 1)
+      assert.strictEqual(compare(-Infinity, 7), -1)
+      assert(isNaN(compare(NaN, 0)))
+      assert(isNaN(compare(false, NaN)))
+      assert(isNaN(compare(NaN, NaN)))
+      assert.strictEqual(compare(true, false), 1)
+      assert.throws(() => compare(undefined, -1), ResolutionError)
+   })
 })
diff --git a/src/generic/__test__/utils.spec.js b/src/generic/__test__/utils.spec.js
new file mode 100644
index 0000000..e178a2d
--- /dev/null
+++ b/src/generic/__test__/utils.spec.js
@@ -0,0 +1,13 @@
+import assert from 'assert'
+import math from '#nanomath'
+
+describe('generic utility functions', () => {
+   it('tests whether an element is zero', () => {
+      const {isZero} = math
+      assert(!isZero(3))
+      assert(isZero(3e-16))
+      assert(isZero(false))
+      assert(!isZero(true))
+      assert(isZero(undefined))
+   })
+})
diff --git a/src/generic/all.js b/src/generic/all.js
index 951d1f6..4dea4d2 100644
--- a/src/generic/all.js
+++ b/src/generic/all.js
@@ -1,3 +1,4 @@
 export * as arithmetic from './arithmetic.js'
 export * as configuration from './config.js'
 export * as relational from './relational.js'
+export * as utilities from './utils.js'
diff --git a/src/generic/relational.js b/src/generic/relational.js
index 7e7f92f..acade73 100644
--- a/src/generic/relational.js
+++ b/src/generic/relational.js
@@ -1,36 +1,52 @@
 import {ReturnsAs} from './helpers.js'
 import {onType} from '#core/helpers.js'
+import {Returns} from '#core/Type.js'
 import {Any, matched} from '#core/TypePatterns.js'
 import {boolnum} from '#number/helpers.js'
+import {NumberT} from '#number/NumberT.js'
 
-export const equal = onType(
-   [Any, Any], (math, [T, U]) => {
-      // Finding the correct signature of `indistinguishable` to use for
-      // testing (approximate) equality is tricky, because T or U might
-      // need to be converted for the sake of comparison, and some types
-      // allow tolerances for equality and others don't. So the plan is
-      // we first look up without tolerances, then we check the config for
-      // the matching type, and then we look up with tolerances.
-      let exactChecker
+export const equal = onType([Any, Any], (math, [T, U]) => {
+   // Finding the correct signature of `indistinguishable` to use for
+   // testing (approximate) equality is tricky, because T or U might
+   // need to be converted for the sake of comparison, and some types
+   // allow tolerances for equality and others don't. So the plan is
+   // we first look up without tolerances, then we check the config for
+   // the matching type, and then we look up with tolerances.
+   let exactChecker
+   try {
+      exactChecker = math.indistinguishable.resolve([T, U])
+   } catch { // can't compare, so no way they can be equal
+      return boolnum(() => false)(math)
+   }
+   // Get the type of the first argument to the matching checker:
+   const ByType = matched(exactChecker.template).flat()[0]
+   // Now see if there are tolerances for that type:
+   const typeConfig = math.resolve('config', [ByType])
+   if ('relTol' in typeConfig) {
       try {
-         exactChecker = math.indistinguishable.resolve([T, U])
-      } catch { // can't compare, so no way they can be equal
-         return boolnum(() => false)(math)
-      }
-      // Get the type of the first argument to the matching checker:
-      const ByType = matched(exactChecker.template).flat()[0]
-      // Now see if there are tolerances for that type:
-      const typeConfig = math.resolve('config', [ByType])
-      if ('relTol' in typeConfig) {
-         try {
-            const {relTol, absTol} = typeConfig
-            const RT = math.typeOf(relTol)
-            const AT = math.typeOf(absTol)
-            const approx = math.indistinguishable.resolve([T, U, RT, AT])
-            return ReturnsAs(
-               approx, (t, u) => approx(t, u, relTol, absTol))
-         } catch {} // fall through to case with no tolerances
-      }
-      // either no tolerances or no matching signature for indistinguishable
-      return exactChecker
-  })
+         const {relTol, absTol} = typeConfig
+         const RT = math.typeOf(relTol)
+         const AT = math.typeOf(absTol)
+         const approx = math.indistinguishable.resolve([T, U, RT, AT])
+         return ReturnsAs(
+            approx, (t, u) => approx(t, u, relTol, absTol))
+      } catch {} // fall through to case with no tolerances
+   }
+   // either no tolerances or no matching signature for indistinguishable
+   return exactChecker
+})
+
+// now that we have `equal` and `exceeds`, pretty much everything else should
+// be easy:
+
+export const compare = onType([Any, Any], (math, [T, U]) => {
+   const eq = math.equal.resolve([T, U])
+   const gt = math.exceeds.resolve([T, U])
+   const isTnan = math.isnan.resolve([T])
+   const isUnan = math.isnan.resolve([U])
+   return Returns(NumberT, (t, u) => {
+      if (isTnan(t) || isUnan(u)) return NaN
+      if (eq(t,u)) return 0
+      return gt(t, u) ? 1 : -1
+   })
+})
diff --git a/src/generic/utils.js b/src/generic/utils.js
new file mode 100644
index 0000000..fc78903
--- /dev/null
+++ b/src/generic/utils.js
@@ -0,0 +1,10 @@
+import {ReturnsAs} from './helpers.js'
+import {onType} from '#core/helpers.js'
+import {Returns} from '#core/Type.js'
+import {Any} from "#core/TypePatterns.js"
+
+export const isZero = (math, [T]) => {
+   const z = math.zero(T)
+   const eq = math.equal.resolve([T, T])
+   return ReturnsAs(eq, x => eq(z, x))
+}
diff --git a/src/number/NumberT.js b/src/number/NumberT.js
index 99c53ff..332fab6 100644
--- a/src/number/NumberT.js
+++ b/src/number/NumberT.js
@@ -4,4 +4,6 @@ import {BooleanT} from '#boolean/BooleanT.js'
 
 export const NumberT = new Type(n => typeof n === 'number', {
     from: onType(BooleanT, math => math.number.resolve([BooleanT])),
+    one: 1,
+    zero: 0
 })
diff --git a/src/number/__test__/relational.spec.js b/src/number/__test__/relational.spec.js
index 9908482..ce5d8eb 100644
--- a/src/number/__test__/relational.spec.js
+++ b/src/number/__test__/relational.spec.js
@@ -12,6 +12,7 @@ describe('number relational functions', () => {
       assert(exceeds(-1e101, -Infinity))
       assert(!exceeds(NaN, 0))
       assert(!exceeds(0, NaN))
+      assert(!exceeds(NaN, NaN))
       assert(!exceeds(2, 2))
    })
    it('checks for exact equality', () => {
diff --git a/src/number/type.js b/src/number/type.js
index 3a9ebfc..9ec8fed 100644
--- a/src/number/type.js
+++ b/src/number/type.js
@@ -7,6 +7,7 @@ const num = f => Returns(NumberT, f)
 
 export const number = plain(a => a)
 number.also(
-   BooleanT, num(p => p ? 1 : 0),
+   // conversions from Boolean should be consistent with one and zero:
+   BooleanT, num(p => p ? NumberT.one : NumberT.zero),
    [], num(() => 0)
 )