diff --git a/bench/src/main/scala/cats/bench/FoldBench.scala b/bench/src/main/scala/cats/bench/FoldBench.scala
index d6388e868e..79021de2c4 100644
--- a/bench/src/main/scala/cats/bench/FoldBench.scala
+++ b/bench/src/main/scala/cats/bench/FoldBench.scala
@@ -1,6 +1,6 @@
 package cats.bench
 
-import algebra.std.string._
+import cats.std.string._
 import cats.data.Const
 import cats.std.list._
 import cats.{Foldable, Traverse}
diff --git a/build.sbt b/build.sbt
index 8ca3ec076a..b0bdff885e 100644
--- a/build.sbt
+++ b/build.sbt
@@ -1,6 +1,8 @@
 import com.typesafe.sbt.pgp.PgpKeys.publishSigned
 import com.typesafe.sbt.SbtSite.SiteKeys._
 import com.typesafe.sbt.SbtGhPages.GhPagesKeys._
+import com.typesafe.tools.mima.plugin.MimaPlugin.mimaDefaultSettings
+import com.typesafe.tools.mima.plugin.MimaKeys.previousArtifacts
 import sbtunidoc.Plugin.UnidocKeys._
 import ReleaseTransformations._
 import ScoverageSbtPlugin._
@@ -23,17 +25,18 @@ lazy val catsDoctestSettings = Seq(
 ) ++ doctestSettings
 
 lazy val commonSettings = Seq(
-  scalacOptions ++= commonScalacOptions,
+  scalacOptions ++= commonScalacOptions ++
+    (CrossVersion.partialVersion(scalaVersion.value) match {
+    case Some((2, 10)) => Seq.empty
+    case _             => Seq("-Ywarn-value-discard")
+  }),
   resolvers ++= Seq(
     "bintray/non" at "http://dl.bintray.com/non/maven",
     Resolver.sonatypeRepo("releases"),
     Resolver.sonatypeRepo("snapshots")
   ),
   libraryDependencies ++= Seq(
-    "com.github.mpilquist" %%% "simulacrum" % "0.6.1",
-    "org.spire-math" %%% "algebra" % "0.3.1",
-    "org.spire-math" %%% "algebra-std" % "0.3.1",
-    "org.typelevel" %%% "machinist" % "0.4.1",
+    "com.github.mpilquist" %%% "simulacrum" % "0.6.1" % "provided",
     compilerPlugin("org.scalamacros" %% "paradise" % "2.1.0-M5" cross CrossVersion.full),
     compilerPlugin("org.spire-math" %% "kind-projector" % "0.6.3")
   ),
@@ -41,6 +44,10 @@ lazy val commonSettings = Seq(
   scalacOptions in (Compile, doc) := (scalacOptions in (Compile, doc)).value.filter(_ != "-Xfatal-warnings")
 ) ++ warnUnusedImport
 
+lazy val machinistDependencies = Seq(
+  libraryDependencies += "org.typelevel" %%% "machinist" % "0.4.1"
+)
+
 lazy val commonJsSettings = Seq(
   scalaJSStage in Global := FastOptStage,
   parallelExecution := false
@@ -52,7 +59,9 @@ lazy val commonJvmSettings = Seq(
 // JVM settings. https://github.com/tkawachi/sbt-doctest/issues/52
 ) ++ catsDoctestSettings
 
-lazy val catsSettings = buildSettings ++ commonSettings ++ publishSettings ++ scoverageSettings
+lazy val kernelSettings = buildSettings ++ commonSettings ++ publishSettings ++ scoverageSettings
+
+lazy val catsSettings = kernelSettings ++ machinistDependencies
 
 lazy val scalacheckVersion = "1.12.5"
 
@@ -63,7 +72,7 @@ lazy val disciplineDependencies = Seq(
 
 lazy val docSettings = Seq(
   autoAPIMappings := true,
-  unidocProjectFilter in (ScalaUnidoc, unidoc) := inProjects(coreJVM),
+  unidocProjectFilter in (ScalaUnidoc, unidoc) := inProjects(kernelJVM, coreJVM),
   site.addMappingsToSiteDir(mappings in (ScalaUnidoc, packageDoc), "api"),
   site.addMappingsToSiteDir(tut, "_tut"),
   ghpagesNoJekyll := false,
@@ -89,7 +98,7 @@ lazy val docs = project
   .settings(tutSettings)
   .settings(tutScalacOptions ~= (_.filterNot(Set("-Ywarn-unused-import", "-Ywarn-dead-code"))))
   .settings(commonJvmSettings)
-  .dependsOn(coreJVM)
+  .dependsOn(kernelJVM, coreJVM)
 
 lazy val cats = project.in(file("."))
   .settings(moduleName := "root")
@@ -102,15 +111,15 @@ lazy val catsJVM = project.in(file(".catsJVM"))
   .settings(moduleName := "cats")
   .settings(catsSettings)
   .settings(commonJvmSettings)
-  .aggregate(macrosJVM, coreJVM, lawsJVM, testsJVM, jvm, docs, bench)
-  .dependsOn(macrosJVM, coreJVM, lawsJVM, testsJVM % "test-internal -> test", jvm, bench % "compile-internal;test-internal -> test")
+  .aggregate(macrosJVM, kernelJVM, coreJVM, lawsJVM, testsJVM, jvm, docs, bench)
+  .dependsOn(macrosJVM, kernelJVM, coreJVM, lawsJVM, testsJVM % "test-internal -> test", jvm, bench % "compile-internal;test-internal -> test")
 
 lazy val catsJS = project.in(file(".catsJS"))
   .settings(moduleName := "cats")
   .settings(catsSettings)
   .settings(commonJsSettings)
-  .aggregate(macrosJS, coreJS, lawsJS, testsJS, js)
-  .dependsOn(macrosJS, coreJS, lawsJS, testsJS % "test-internal -> test", js)
+  .aggregate(macrosJS, kernelJS, coreJS, lawsJS, testsJS, js)
+  .dependsOn(macrosJS, kernelJS, coreJS, lawsJS, testsJS % "test-internal -> test", js)
   .enablePlugins(ScalaJSPlugin)
 
 
@@ -124,9 +133,32 @@ lazy val macros = crossProject.crossType(CrossType.Pure)
 lazy val macrosJVM = macros.jvm
 lazy val macrosJS = macros.js
 
+lazy val kernel = crossProject.crossType(CrossType.Pure)
+  .settings(moduleName := "cats-kernel")
+  .settings(kernelSettings:_*)
+  .settings(mimaDefaultSettings:_*)
+  .settings(previousArtifacts := Set(
+    // TODO: Add cats-kernel artifacts as they are released, e.g.
+    //   "org.spire-math" %% "cats-kernel" % "0.4.0"
+  ))
+  .settings(pomPostProcess := { (node) =>
+    import scala.xml._
+    import scala.xml.transform._
+    def stripIf(f: Node => Boolean) = new RewriteRule {
+      override def transform(n: Node) =
+        if (f(n)) NodeSeq.Empty else n
+    }
+    val stripProvidedScope = stripIf { n => n.label == "dependency" && (n \ "scope").text == "provided" }
+    new RuleTransformer(stripProvidedScope).transform(node)(0)
+  })
+  .jsSettings(commonJsSettings:_*)
+  .jvmSettings(commonJvmSettings:_*)
+
+lazy val kernelJVM = kernel.jvm
+lazy val kernelJS = kernel.js
 
 lazy val core = crossProject.crossType(CrossType.Pure)
-  .dependsOn(macros)
+  .dependsOn(macros, kernel)
   .settings(moduleName := "cats-core")
   .settings(catsSettings:_*)
   .settings(
@@ -140,12 +172,11 @@ lazy val coreJVM = core.jvm
 lazy val coreJS = core.js
 
 lazy val laws = crossProject.crossType(CrossType.Pure)
-  .dependsOn(macros, core)
+  .dependsOn(macros, kernel, core)
   .settings(moduleName := "cats-laws")
   .settings(catsSettings:_*)
   .settings(disciplineDependencies:_*)
   .settings(libraryDependencies ++= Seq(
-    "org.spire-math" %%% "algebra-laws" % "0.3.1",
     "org.typelevel" %%% "catalysts-platform" % "0.0.2"))
   .jsSettings(commonJsSettings:_*)
   .jvmSettings(commonJvmSettings:_*)
@@ -154,7 +185,7 @@ lazy val lawsJVM = laws.jvm
 lazy val lawsJS = laws.js
 
 lazy val tests = crossProject.crossType(CrossType.Pure)
-  .dependsOn(macros, core, laws)
+  .dependsOn(macros, kernel, core, laws)
   .settings(moduleName := "cats-tests")
   .settings(catsSettings:_*)
   .settings(disciplineDependencies:_*)
@@ -170,13 +201,13 @@ lazy val testsJS = tests.js
 
 // cats-jvm is JVM-only
 lazy val jvm = project
-  .dependsOn(macrosJVM, coreJVM, testsJVM % "test-internal -> test")
+  .dependsOn(macrosJVM, kernelJVM, coreJVM, testsJVM % "test-internal -> test")
   .settings(moduleName := "cats-jvm")
   .settings(catsSettings:_*)
   .settings(commonJvmSettings:_*)
 
 // bench is currently JVM-only
-lazy val bench = project.dependsOn(macrosJVM, coreJVM, lawsJVM)
+lazy val bench = project.dependsOn(macrosJVM, kernelJVM, coreJVM, lawsJVM)
   .settings(moduleName := "cats-bench")
   .settings(catsSettings)
   .settings(noPublishSettings)
@@ -185,7 +216,7 @@ lazy val bench = project.dependsOn(macrosJVM, coreJVM, lawsJVM)
 
 // cats-js is JS-only
 lazy val js = project
-  .dependsOn(macrosJS, coreJS, testsJS % "test-internal -> test")
+  .dependsOn(macrosJS, kernelJS, coreJS, testsJS % "test-internal -> test")
   .settings(moduleName := "cats-js")
   .settings(catsSettings:_*)
   .settings(commonJsSettings:_*)
@@ -209,11 +240,11 @@ lazy val publishSettings = Seq(
 ) ++ credentialSettings ++ sharedPublishSettings ++ sharedReleaseProcess 
 
 // These aliases serialise the build for the benefit of Travis-CI.
-addCommandAlias("buildJVM", ";macrosJVM/compile;coreJVM/compile;coreJVM/test;lawsJVM/compile;testsJVM/test;jvm/test;bench/test")
+addCommandAlias("buildJVM", ";macrosJVM/compile;kernelJVM/compile;coreJVM/compile;kernelJVM/test;coreJVM/test;lawsJVM/compile;testsJVM/test;jvm/test;bench/test")
 
-addCommandAlias("validateJVM", ";scalastyle;buildJVM;makeSite")
+addCommandAlias("validateJVM", ";scalastyle;buildJVM;kernelJVM/mima-report-binary-issues;makeSite")
 
-addCommandAlias("validateJS", ";macrosJS/compile;coreJS/compile;lawsJS/compile;testsJS/test;js/test")
+addCommandAlias("validateJS", ";macrosJS/compile;kernelJS/compile;coreJS/compile;lawsJS/compile;testsJS/test;js/test")
 
 addCommandAlias("validate", ";validateJS;validateJVM")
 
@@ -271,7 +302,6 @@ lazy val commonScalacOptions = Seq(
   "-Yno-adapted-args",
   "-Ywarn-dead-code",
   "-Ywarn-numeric-widen",
-  "-Ywarn-value-discard",
   "-Xfuture"
 )
 
diff --git a/core/src/main/scala/cats/package.scala b/core/src/main/scala/cats/package.scala
index 00c575017b..c55006c772 100644
--- a/core/src/main/scala/cats/package.scala
+++ b/core/src/main/scala/cats/package.scala
@@ -44,18 +44,4 @@ package object cats {
       def traverse[G[_], A, B](a: A)(f: A => G[B])(implicit G: Applicative[G]): G[B] =
         f(a)
   }
-
-  type Eq[A] = algebra.Eq[A]
-  type PartialOrder[A] = algebra.PartialOrder[A]
-  type Order[A] = algebra.Order[A]
-  type Semigroup[A] = algebra.Semigroup[A]
-  type Monoid[A] = algebra.Monoid[A]
-  type Group[A] = algebra.Group[A]
-
-  val Eq = algebra.Eq
-  val PartialOrder = algebra.PartialOrder
-  val Order = algebra.Order
-  val Semigroup = algebra.Semigroup
-  val Monoid = algebra.Monoid
-  val Group = algebra.Group
 }
diff --git a/core/src/main/scala/cats/std/anyval.scala b/core/src/main/scala/cats/std/anyval.scala
index 7964aa18e9..9c11fd2cc6 100644
--- a/core/src/main/scala/cats/std/anyval.scala
+++ b/core/src/main/scala/cats/std/anyval.scala
@@ -1,9 +1,6 @@
 package cats
 package std
 
-import algebra.CommutativeGroup
-import algebra.ring.AdditiveCommutativeGroup
-
 trait AnyValInstances
   extends IntInstances
   with    ByteInstances
@@ -15,33 +12,35 @@ trait AnyValInstances
   with    BooleanInstances
   with    UnitInstances
 
-trait IntInstances extends algebra.std.IntInstances {
+trait IntInstances {
 
   implicit val intShow: Show[Int] =
     Show.fromToString[Int]
 
-  implicit val intGroup: CommutativeGroup[Int] =
-    AdditiveCommutativeGroup[Int].additive
+  implicit val intAlgebra: Group[Int] with Order[Int] =
+    new Group[Int] with Order[Int] {
+      def empty: Int = 0
+      def inverse(a: Int): Int = -a
+      def combine(x: Int, y: Int): Int = x+y
+      def compare(x: Int, y: Int): Int =
+        if (x < y) -1 else if (x > y) 1 else 0
+    }
 
 }
 
-trait ByteInstances /* missing algebra type classes */ {
+trait ByteInstances {
 
   implicit val byteShow: Show[Byte] =
     Show.fromToString[Byte]
 
-  // TODO: replace this minimal algebra with one from the algebra project
-  implicit val byteAlgebra: CommutativeGroup[Byte] with Order[Byte] =
-    new CommutativeGroup[Byte] with Order[Byte] {
-      def combine(x: Byte, y: Byte): Byte = (x + y).toByte
-      def empty: Byte = 0
-      def inverse(x: Byte): Byte = (-x).toByte
+  implicit val byteOrder: Order[Byte] =
+    new Order[Byte] {
       def compare(x: Byte, y: Byte): Int =
         if (x < y) -1 else if (y < x) 1 else 0
     }
 }
 
-trait CharInstances /* missing algebra type classes */ {
+trait CharInstances {
 
   implicit val charShow: Show[Char] =
     Show.fromToString[Char]
@@ -53,90 +52,80 @@ trait CharInstances /* missing algebra type classes */ {
     }
 }
 
-trait ShortInstances /* missing algebra type classes */ {
+trait ShortInstances {
 
   implicit val shortShow: Show[Short] =
     Show.fromToString[Short]
 
-  // TODO: replace this minimal algebra with one from the algebra project
-  implicit val shortAlgebra: CommutativeGroup[Short] with Order[Short] =
-    new CommutativeGroup[Short] with Order[Short] {
-      def combine(x: Short, y: Short): Short = (x + y).toShort
-      def empty: Short = 0
-      def inverse(x: Short): Short = (-x).toShort
+  implicit val shortOrder: Order[Short] =
+    new Order[Short] {
       def compare(x: Short, y: Short): Int =
         if (x < y) -1 else if (y < x) 1 else 0
     }
 
 }
 
-trait LongInstances /* missing algebra type classes */ {
+trait LongInstances {
 
   implicit val longShow: Show[Long] =
     Show.fromToString[Long]
 
-  // TODO: replace this minimal algebra with one from the algebra project
-  implicit val longAlgebra: CommutativeGroup[Long] with Order[Long] =
-    new CommutativeGroup[Long] with Order[Long] {
-      def combine(x: Long, y: Long): Long = x + y
+  implicit val longAlgebra: Group[Long] with Order[Long] =
+    new Group[Long] with Order[Long] {
       def empty: Long = 0L
-      def inverse(x: Long): Long = -x
+      def inverse(a: Long): Long = -a
+      def combine(x: Long, y: Long): Long = x+y
       def compare(x: Long, y: Long): Int =
         if (x < y) -1 else if (y < x) 1 else 0
     }
 }
 
-trait FloatInstances /* missing algebra type classes */ {
+trait FloatInstances {
 
   implicit val floatShow: Show[Float] =
     Show.fromToString[Float]
 
-  // TODO: replace this minimal algebra with one from the algebra project
-  implicit val floatAlgebra: CommutativeGroup[Float] with Order[Float] =
-    new CommutativeGroup[Float] with Order[Float] {
-      def combine(x: Float, y: Float): Float = x + y
-      def empty: Float = 0F
-      def inverse(x: Float): Float = -x
+  implicit val floatOrder: Order[Float] =
+    new Order[Float] {
       def compare(x: Float, y: Float): Int =
         java.lang.Float.compare(x, y)
     }
 
 }
 
-trait DoubleInstances /* missing algebra type classes */ {
+trait DoubleInstances {
 
   implicit val doubleShow: Show[Double] =
     Show.fromToString[Double]
 
-  // TODO: replace this minimal algebra with one from the algebra project
-  implicit val doubleAlgebra: CommutativeGroup[Double] with Order[Double] =
-    new CommutativeGroup[Double] with Order[Double] {
-      def combine(x: Double, y: Double): Double = x + y
-      def empty: Double = 0D
-      def inverse(x: Double): Double = -x
+  implicit val doubleOrder: Order[Double] =
+    new Order[Double] {
       def compare(x: Double, y: Double): Int =
         java.lang.Double.compare(x, y)
     }
 
 }
 
-trait BooleanInstances extends algebra.std.BooleanInstances {
+trait BooleanInstances {
 
   implicit val booleanShow: Show[Boolean] =
     Show.fromToString[Boolean]
 
+  implicit val booleanOrder: Order[Boolean] =
+    new Order[Boolean] {
+      def compare(x: Boolean, y: Boolean): Int =
+        if (x == y) 0 else if (x) 1 else -1
+    }
+
 }
 
-trait UnitInstances /* missing algebra type classes */ {
+trait UnitInstances {
 
   implicit val unitShow: Show[Unit] =
     Show.fromToString[Unit]
 
-  implicit val unitAlgebra: CommutativeGroup[Unit] with Order[Unit] =
-    new CommutativeGroup[Unit] with Order[Unit] {
-      def combine(x: Unit, y: Unit): Unit = ()
-      def empty: Unit = ()
-      def inverse(x: Unit): Unit = ()
+  implicit val unitOrder: Order[Unit] =
+    new Order[Unit] {
       def compare(x: Unit, y: Unit): Int = 0
     }
 
diff --git a/core/src/main/scala/cats/std/bigInt.scala b/core/src/main/scala/cats/std/bigInt.scala
index d075af3826..98c471f832 100644
--- a/core/src/main/scala/cats/std/bigInt.scala
+++ b/core/src/main/scala/cats/std/bigInt.scala
@@ -1,7 +1,14 @@
 package cats
 package std
 
-trait BigIntInstances extends algebra.std.BigIntInstances {
+trait BigIntInstances {
+  implicit val bigIntAlgebra: BigIntAlgebra =
+    new BigIntAlgebra
+
   implicit val bigIntShow: Show[BigInt] =
     Show.fromToString[BigInt]
 }
+
+class BigIntAlgebra extends Order[BigInt] {
+  def compare(x: BigInt, y: BigInt): Int = x compare y
+}
diff --git a/core/src/main/scala/cats/std/function.scala b/core/src/main/scala/cats/std/function.scala
index 8d7c3af2d7..e485c44c35 100644
--- a/core/src/main/scala/cats/std/function.scala
+++ b/core/src/main/scala/cats/std/function.scala
@@ -1,7 +1,6 @@
 package cats
 package std
 
-import algebra.Eq
 import cats.arrow.{Arrow, Choice}
 import cats.data.Xor
 import cats.functor.Contravariant
diff --git a/core/src/main/scala/cats/std/list.scala b/core/src/main/scala/cats/std/list.scala
index 69575cf14e..3af0b3c619 100644
--- a/core/src/main/scala/cats/std/list.scala
+++ b/core/src/main/scala/cats/std/list.scala
@@ -1,9 +1,6 @@
 package cats
 package std
 
-import algebra.Eq
-import algebra.std.{ListMonoid, ListOrder}
-
 import cats.data.Streaming
 import cats.syntax.order._
 import cats.syntax.show._
@@ -120,3 +117,42 @@ private[std] sealed trait ListInstances2 {
       }
     }
 }
+
+class ListOrder[A](implicit ev: Order[A]) extends Order[List[A]] {
+  def compare(xs: List[A], ys: List[A]): Int = {
+    @tailrec def loop(xs: List[A], ys: List[A]): Int =
+      xs match {
+        case Nil =>
+          if (ys.isEmpty) 0 else -1
+        case x :: xs =>
+          ys match {
+            case Nil => 1
+            case y :: ys =>
+              val n = ev.compare(x, y)
+              if (n != 0) n else loop(xs, ys)
+          }
+      }
+    loop(xs, ys)
+  }
+}
+
+class ListMonoid[A] extends Monoid[List[A]] {
+  def empty: List[A] = Nil
+  def combine(x: List[A], y: List[A]): List[A] = x ::: y
+
+  override def combineN(x: List[A], n: Int): List[A] = {
+    val buf = ListBuffer.empty[A]
+    @tailrec def loop(i: Int): List[A] =
+      if (i <= 0) buf.toList else {
+        buf ++= x
+        loop(i - 1)
+      }
+    loop(n)
+  }
+
+  override def combineAll(xs: TraversableOnce[List[A]]): List[A] = {
+    val buf = ListBuffer.empty[A]
+    xs.foreach(buf ++= _)
+    buf.toList
+  }
+}
diff --git a/core/src/main/scala/cats/std/map.scala b/core/src/main/scala/cats/std/map.scala
index bb6e2b35e0..101f72d028 100644
--- a/core/src/main/scala/cats/std/map.scala
+++ b/core/src/main/scala/cats/std/map.scala
@@ -1,9 +1,11 @@
 package cats
 package std
 
+import scala.collection.mutable
+
 import cats.syntax.eq._
 
-trait MapInstances extends algebra.std.MapInstances {
+trait MapInstances {
 
   implicit def MapEq[A, B: Eq]: Eq[Map[A, B]] =
     new Eq[Map[A, B]] {
@@ -64,4 +66,72 @@ trait MapInstances extends algebra.std.MapInstances {
 
       override def isEmpty[A](fa: Map[K, A]): Boolean = fa.isEmpty
     }
+
+  implicit def mapMonoid[K, V: Semigroup]: MapMonoid[K, V] =
+    new MapMonoid[K, V]
+
+  implicit def mapGroup[K, V: Group]: MapGroup[K, V] =
+    new MapGroup[K, V]
+}
+
+class MapMonoid[K, V](implicit V: Semigroup[V]) extends Monoid[Map[K, V]]  {
+  def empty: Map[K, V] = Map.empty
+
+  def combine(x: Map[K, V], y: Map[K, V]): Map[K, V] =
+    MapMethods.addMap(x, y)(V.combine)
+}
+
+class MapGroup[K, V](implicit V: Group[V]) extends MapMonoid[K, V] with Group[Map[K, V]] {
+  def inverse(x: Map[K, V]): Map[K, V] =
+    x.map { case (k, v) => (k, V.inverse(v)) }
+
+  override def remove(x: Map[K, V], y: Map[K, V]): Map[K, V] =
+    MapMethods.subtractMap(x, y)(V.remove)(V.inverse)
+}
+
+object MapMethods {
+  def initMutableMap[K, V](m: Map[K, V]): mutable.Map[K, V] = {
+    val result = mutable.Map.empty[K, V]
+    m.foreach { case (k, v) => result(k) = v }
+    result
+  }
+
+  def wrapMutableMap[K, V](m: mutable.Map[K, V]): Map[K, V] =
+    new WrappedMutableMap(m)
+
+  private[cats] class WrappedMutableMap[K, V](m: mutable.Map[K, V]) extends Map[K, V] {
+    override def size: Int = m.size
+    def get(k: K): Option[V] = m.get(k)
+    def iterator: Iterator[(K, V)] = m.iterator
+    def +[V2 >: V](kv: (K, V2)): Map[K, V2] = m.toMap + kv
+    def -(key: K): Map[K, V] = m.toMap - key
+  }
+
+  def addMap[K, V](x: Map[K, V], y: Map[K, V])(f: (V, V) => V): Map[K, V] = {
+    val (small, big, g) =
+      if (x.size <= y.size) (x, y, f)
+      else (y, x, (v1: V, v2: V) => f(v2, v1))
+
+    val m = initMutableMap(big)
+    small.foreach { case (k, v1) =>
+      m(k) = m.get(k) match {
+        case Some(v2) => g(v1, v2)
+        case None => v1
+      }
+    }
+    wrapMutableMap(m)
+  }
+
+  def subtractMap[K, V](x: Map[K, V], y: Map[K, V])(subtract: (V, V) => V)(negate: V => V): Map[K, V] = {
+    // even if x is smaller, we'd need to call map/foreach on y to
+    // negate all its values, so this is just as fast or faster.
+    val m = initMutableMap(x)
+    y.foreach { case (k, v2) =>
+      m(k) = m.get(k) match {
+        case Some(v1) => subtract(v1, v2)
+        case None => negate(v2)
+      }
+    }
+    wrapMutableMap(m)
+  }
 }
diff --git a/core/src/main/scala/cats/std/option.scala b/core/src/main/scala/cats/std/option.scala
index 4439a4e0a0..dcaeee56e7 100644
--- a/core/src/main/scala/cats/std/option.scala
+++ b/core/src/main/scala/cats/std/option.scala
@@ -1,8 +1,6 @@
 package cats
 package std
 
-import algebra.Eq
-
 trait OptionInstances extends OptionInstances1 {
   implicit val optionInstance: Traverse[Option] with MonadCombine[Option] with CoflatMap[Option] with Alternative[Option] =
     new Traverse[Option] with MonadCombine[Option] with CoflatMap[Option] with Alternative[Option] {
diff --git a/core/src/main/scala/cats/std/set.scala b/core/src/main/scala/cats/std/set.scala
index 1efeff8350..4849ef43d4 100644
--- a/core/src/main/scala/cats/std/set.scala
+++ b/core/src/main/scala/cats/std/set.scala
@@ -3,7 +3,7 @@ package std
 
 import cats.syntax.show._
 
-trait SetInstances extends algebra.std.SetInstances {
+trait SetInstances {
   implicit val setInstance: Foldable[Set] with MonoidK[Set] =
     new Foldable[Set] with MonoidK[Set] {
 
@@ -28,8 +28,20 @@ trait SetInstances extends algebra.std.SetInstances {
 
     implicit def setMonoid[A]: Monoid[Set[A]] = MonoidK[Set].algebra[A]
 
+  implicit def setPartialOrder[A]: PartialOrder[Set[A]] = new SetPartialOrder[A]
+
   implicit def setShow[A:Show]: Show[Set[A]] = new Show[Set[A]] {
     def show(fa: Set[A]): String =
       fa.toIterator.map(_.show).mkString("Set(", ", ", ")")
   }
 }
+
+class SetPartialOrder[A] extends PartialOrder[Set[A]] {
+  def partialCompare(x: Set[A], y: Set[A]): Double =
+    if (x.size < y.size) if (x.subsetOf(y)) -1.0 else Double.NaN
+    else if (y.size < x.size) -partialCompare(y, x)
+    else if (x == y) 0.0
+    else Double.NaN
+
+  override def eqv(x: Set[A], y: Set[A]): Boolean = x == y
+}
diff --git a/core/src/main/scala/cats/std/string.scala b/core/src/main/scala/cats/std/string.scala
index a79a814734..a7b5cb2200 100644
--- a/core/src/main/scala/cats/std/string.scala
+++ b/core/src/main/scala/cats/std/string.scala
@@ -1,7 +1,24 @@
 package cats
 package std
 
-trait StringInstances extends algebra.std.StringInstances {
+trait StringInstances {
+  implicit val stringOrder: Order[String] = new StringOrder
+  implicit val stringMonoid = new StringMonoid
   implicit val stringShow: Show[String] =
     Show.fromToString[String]
 }
+
+class StringOrder extends Order[String] {
+  def compare(x: String, y: String): Int = x compare y
+}
+
+class StringMonoid extends Monoid[String] {
+  def empty: String = ""
+  def combine(x: String, y: String): String = x + y
+
+  override def combineAll(xs: TraversableOnce[String]): String = {
+    val sb = new StringBuilder
+    xs.foreach(sb.append)
+    sb.toString
+  }
+}
diff --git a/docs/src/main/tut/eq.md b/docs/src/main/tut/eq.md
new file mode 100644
index 0000000000..61a580cb54
--- /dev/null
+++ b/docs/src/main/tut/eq.md
@@ -0,0 +1,9 @@
+---
+layout: default
+title:  "Eq"
+section: "typeclasses"
+source: "https://github.com/non/cats/blob/master/kernel/src/main/scala/cats/Eq.scala"
+scaladoc: "#cats.Eq"
+---
+# Eq
+
diff --git a/docs/src/main/tut/group.md b/docs/src/main/tut/group.md
new file mode 100644
index 0000000000..043e27b3bb
--- /dev/null
+++ b/docs/src/main/tut/group.md
@@ -0,0 +1,9 @@
+---
+layout: default
+title:  "Group"
+section: "typeclasses"
+source: "https://github.com/non/cats/blob/master/kernel/src/main/scala/cats/Group.scala"
+scaladoc: "#cats.Group"
+---
+# Group
+
diff --git a/docs/src/main/tut/monoid.md b/docs/src/main/tut/monoid.md
index e9f9b6020f..b1b6fe1c9e 100644
--- a/docs/src/main/tut/monoid.md
+++ b/docs/src/main/tut/monoid.md
@@ -2,7 +2,7 @@
 layout: default
 title:  "Monoid"
 section: "typeclasses"
-source: "https://github.com/non/algebra/blob/master/core/src/main/scala/algebra/Monoid.scala"
+source: "https://github.com/non/cats/blob/master/kernel/src/main/scala/cats/Monoid.scala"
 
 ---
 # Monoid
@@ -81,13 +81,3 @@ Thus.
 ```tut
 l.foldMap(i => (i, i.toString)) // do both of the above in one pass, hurrah!
 ```
-
--------------------------------------------------------------------------------
- 
-N.B.
-Cats does not define a `Monoid` type class itself, it uses the [`Monoid`
-trait](https://github.com/non/algebra/blob/master/core/src/main/scala/algebra/Monoid.scala)
-which is defined in the [algebra project](https://github.com/non/algebra) on 
-which it depends. The [`cats` package object](https://github.com/non/cats/blob/master/core/src/main/scala/cats/package.scala)
-defines type aliases to the `Monoid` from algebra, so that you can
-`import cats.Monoid`.
diff --git a/docs/src/main/tut/order.md b/docs/src/main/tut/order.md
new file mode 100644
index 0000000000..f79a9d295f
--- /dev/null
+++ b/docs/src/main/tut/order.md
@@ -0,0 +1,9 @@
+---
+layout: default
+title:  "Order"
+section: "typeclasses"
+source: "https://github.com/non/cats/blob/master/kernel/src/main/scala/cats/Order.scala"
+scaladoc: "#cats.Order"
+---
+# Order
+
diff --git a/docs/src/main/tut/partialorder.md b/docs/src/main/tut/partialorder.md
new file mode 100644
index 0000000000..8e0c9894cd
--- /dev/null
+++ b/docs/src/main/tut/partialorder.md
@@ -0,0 +1,9 @@
+---
+layout: default
+title:  "PartialOrder"
+section: "typeclasses"
+source: "https://github.com/non/cats/blob/master/kernel/src/main/scala/cats/PartialOrder.scala"
+scaladoc: "#cats.PartialOrder"
+---
+# PartialOrder
+
diff --git a/docs/src/main/tut/semigroup.md b/docs/src/main/tut/semigroup.md
index 3dad3715cd..75d9bbd1ba 100644
--- a/docs/src/main/tut/semigroup.md
+++ b/docs/src/main/tut/semigroup.md
@@ -2,7 +2,7 @@
 layout: default
 title:  "Semigroup"
 section: "typeclasses"
-source: "https://github.com/non/algebra/blob/master/core/src/main/scala/algebra/Semigroup.scala"
+source: "https://github.com/non/cats/blob/master/kernel/src/main/scala/cats/Semigroup.scala"
 
 ---
 # Semigroup
@@ -94,11 +94,3 @@ Option. If we try to use Some and None, we'll get errors:
 Some(1) |+| None
 None |+| Some(1)
 ```
-
-N.B.
-Cats does not define a `Semigroup` type class itself, it uses the [`Semigroup`
-trait](https://github.com/non/algebra/blob/master/core/src/main/scala/algebra/Semigroup.scala)
-which is defined in the [algebra project](https://github.com/non/algebra) on 
-which it depends. The [`cats` package object](https://github.com/non/cats/blob/master/core/src/main/scala/cats/package.scala)
-defines type aliases to the `Semigroup` from algebra, so that you can
-`import cats.Semigroup`.
\ No newline at end of file
diff --git a/kernel/src/main/scala/cats/Eq.scala b/kernel/src/main/scala/cats/Eq.scala
new file mode 100644
index 0000000000..467bd899cd
--- /dev/null
+++ b/kernel/src/main/scala/cats/Eq.scala
@@ -0,0 +1,103 @@
+package cats
+
+import simulacrum.typeclass
+
+import scala.{specialized => sp}
+
+import scala.math.Equiv
+
+/**
+ * A type class used to determine equality between 2 instances of the same
+ * type. Any 2 instances `x` and `y` are equal if `eqv(x, y)` is `true`.
+ * Moreover, `eqv` should form an equivalence relation.
+ */
+@typeclass trait Eq[@sp A] { self =>
+
+  /**
+   * Returns `true` if `x` and `y` are equivalent, `false` otherwise.
+   */
+  def eqv(x: A, y: A): Boolean
+
+  /**
+   * Returns `false` if `x` and `y` are equivalent, `true` otherwise.
+   */
+  def neqv(x: A, y: A): Boolean = !eqv(x, y)
+
+  /**
+   * Constructs a new `Eq` instance for type `B` where 2 elements are
+   * equivalent iff `eqv(f(x), f(y))`.
+   */
+  def on[@sp B](f: B => A): Eq[B] =
+    new Eq[B] {
+      def eqv(x: B, y: B): Boolean = self.eqv(f(x), f(y))
+    }
+
+  /**
+   * Return an Eq that gives the result of the and of this and that
+   * note this is idempotent
+   */
+  def and(that: Eq[A]): Eq[A] =
+    new Eq[A] {
+      def eqv(x: A, y: A) = self.eqv(x, y) && that.eqv(x, y)
+    }
+  /**
+   * Return an Eq that gives the result of the or of this and that
+   * Note this is idempotent
+   */
+  def or(that: Eq[A]): Eq[A] =
+    new Eq[A] {
+      def eqv(x: A, y: A) = self.eqv(x, y) || that.eqv(x, y)
+    }
+}
+
+trait EqFunctions {
+  def eqv[@sp A](x: A, y: A)(implicit ev: Eq[A]): Boolean =
+    ev.eqv(x, y)
+
+  def neqv[@sp A](x: A, y: A)(implicit ev: Eq[A]): Boolean =
+    ev.neqv(x, y)
+}
+
+object Eq extends EqFunctions {
+
+  /**
+   * Convert an implicit `Eq[B]` to an `Eq[A]` using the given
+   * function `f`.
+   */
+  def by[@sp A, @sp B](f: A => B)(implicit ev: Eq[B]): Eq[A] =
+    ev.on(f)
+
+  /**
+   * This gives compatibility with scala's Equiv trait
+   */
+  implicit def equiv[A](implicit ev: Eq[A]): Equiv[A] =
+    new Equiv[A] {
+      def equiv(a: A, b: A) = ev.eqv(a, b)
+    }
+
+  /**
+   * Create an `Eq` instance from an `eqv` implementation.
+   */
+  def instance[A](f: (A, A) => Boolean): Eq[A] =
+    new Eq[A] {
+      def eqv(x: A, y: A) = f(x, y)
+    }
+
+  /**
+   * An `Eq[A]` that delegates to universal equality (`==`).
+   *
+   * This can be useful for case classes, which have reasonable `equals`
+   * implementations
+   */
+  def fromUniversalEquals[A]: Eq[A] =
+    new Eq[A] {
+      def eqv(x: A, y: A) = x == y
+    }
+
+  /**
+   * Everything is the same
+   */
+  def allEqual[A]: Eq[A] = new Eq[A] {
+    def eqv(x: A, y: A) = true
+  }
+}
diff --git a/kernel/src/main/scala/cats/Group.scala b/kernel/src/main/scala/cats/Group.scala
new file mode 100644
index 0000000000..ae38e239cd
--- /dev/null
+++ b/kernel/src/main/scala/cats/Group.scala
@@ -0,0 +1,43 @@
+package cats
+
+import simulacrum.typeclass
+import scala.{ specialized => sp }
+
+/**
+ * A group is a monoid where each element has an inverse.
+ */
+@typeclass trait Group[@sp(Int, Long, Float, Double) A] extends Monoid[A] {
+
+  /**
+   * Find the inverse of `a`.
+   *
+   * `combine(a, inverse(a))` = `combine(inverse(a), a)` = `empty`.
+   */
+  def inverse(a: A): A
+
+  /**
+   * Remove the element `b` from `a`.
+   *
+   * Equivalent to `combine(a, inverse(a))`
+   */
+  def remove(a: A, b: A): A = combine(a, inverse(b))
+
+  /**
+   * Return `a` appended to itself `n` times. If `n` is negative, then
+   * this returns `inverse(a)` appended to itself `n` times.
+   */
+  override def combineN(a: A, n: Int): A =
+    if (n > 0) repeatedCombineN(a, n)
+    else if (n == 0) empty
+    else if (n == Int.MinValue) combineN(inverse(combine(a, a)), 1073741824)
+    else repeatedCombineN(inverse(a), -n)
+}
+
+trait GroupFunctions[G[T] <: Group[T]] extends MonoidFunctions[Group] {
+  def inverse[@sp(Int, Long, Float, Double) A](a: A)(implicit ev: G[A]): A =
+    ev.inverse(a)
+  def remove[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: G[A]): A =
+    ev.remove(x, y)
+}
+
+object Group extends GroupFunctions[Group]
diff --git a/kernel/src/main/scala/cats/Monoid.scala b/kernel/src/main/scala/cats/Monoid.scala
new file mode 100644
index 0000000000..5f1509c17d
--- /dev/null
+++ b/kernel/src/main/scala/cats/Monoid.scala
@@ -0,0 +1,47 @@
+package cats
+
+import simulacrum.typeclass
+import scala.{ specialized => sp }
+
+/**
+ * A monoid is a semigroup with an identity. A monoid is a specialization of a
+ * semigroup, so its operation must be associative. Additionally,
+ * `combine(x, empty) == combine(empty, x) == x`. For example, if we have `Monoid[String]`,
+ * with `combine` as string concatenation, then `empty = ""`.
+ */
+@typeclass trait Monoid[@sp(Int, Long, Float, Double) A] extends Semigroup[A] {
+
+  /**
+   * Return the identity element for this monoid.
+   */
+  def empty: A
+
+  /**
+   * Tests if `a` is the identity.
+   */
+  def isEmpty(a: A)(implicit ev: Eq[A]) = ev.eqv(a, empty)
+
+  /**
+   * Return `a` appended to itself `n` times.
+   */
+  override def combineN(a: A, n: Int): A =
+    if (n < 0) throw new IllegalArgumentException("Repeated combining for monoids must have n >= 0")
+    else if (n == 0) empty
+    else repeatedCombineN(a, n)
+
+  /**
+   * Given a sequence of `as`, sum them using the monoid and return the total.
+   */
+  def combineAll(as: TraversableOnce[A]): A =
+    as.foldLeft(empty)(combine)
+}
+
+trait MonoidFunctions[M[T] <: Monoid[T]] extends SemigroupFunctions[M] {
+  def empty[@sp(Int, Long, Float, Double) A](implicit ev: M[A]): A =
+    ev.empty
+
+  def combineAll[@sp(Int, Long, Float, Double) A](as: TraversableOnce[A])(implicit ev: M[A]): A =
+    ev.combineAll(as)
+}
+
+object Monoid extends MonoidFunctions[Monoid]
diff --git a/kernel/src/main/scala/cats/Order.scala b/kernel/src/main/scala/cats/Order.scala
new file mode 100644
index 0000000000..949c1e7c3d
--- /dev/null
+++ b/kernel/src/main/scala/cats/Order.scala
@@ -0,0 +1,205 @@
+package cats
+
+import simulacrum.typeclass
+import scala.{specialized => sp}
+
+/**
+ * The `Order` type class is used to define a total ordering on some type `A`.
+ * An order is defined by a relation <=, which obeys the following laws:
+ *
+ * - either x <= y or y <= x (totality)
+ * - if x <= y and y <= x, then x == y (antisymmetry)
+ * - if x <= y and y <= z, then x <= z (transitivity)
+ *
+ * The truth table for compare is defined as follows:
+ *
+ * x <= y    x >= y      Int
+ * true      true        = 0     (corresponds to x == y)
+ * true      false       < 0     (corresponds to x < y)
+ * false     true        > 0     (corresponds to x > y)
+ *
+ * By the totality law, x <= y and y <= x cannot be both false.
+ */
+@typeclass trait Order[@sp A] extends PartialOrder[A] { self =>
+
+  /**
+   * Result of comparing `x` with `y`. Returns an Int whose sign is:
+   * - negative iff `x < y`
+   * - zero     iff `x = y`
+   * - positive iff `x > y`
+   */
+  def compare(x: A, y: A): Int
+
+  override def partialCompare(x: A, y: A): Double = compare(x, y).toDouble
+
+  /**
+   * If x <= y, return x, else return y.
+   */
+  def min(x: A, y: A): A = if (lt(x, y)) x else y
+
+  /**
+   * If x >= y, return x, else return y.
+   */
+  def max(x: A, y: A): A = if (gt(x, y)) x else y
+
+  /**
+   * Defines an order on `B` by mapping `B` to `A` using `f` and using `A`s
+   * order to order `B`.
+   */
+  override def on[@sp B](f: B => A): Order[B] =
+    new Order[B] {
+      def compare(x: B, y: B): Int = self.compare(f(x), f(y))
+    }
+
+  /**
+   * Defines an ordering on `A` where all arrows switch direction.
+   */
+  override def reverse: Order[A] =
+    new Order[A] {
+      def compare(x: A, y: A): Int = self.compare(y, x)
+
+      override def reverse: Order[A] = self
+    }
+
+  // The following may be overridden for performance:
+
+  /**
+   * Returns true if `x` = `y`, false otherwise.
+   */
+  override def eqv(x: A, y: A): Boolean =
+    compare(x, y) == 0
+
+  /**
+   * Returns true if `x` != `y`, false otherwise.
+   */
+  override def neqv(x: A, y: A): Boolean =
+    compare(x, y) != 0
+
+  /**
+   * Returns true if `x` <= `y`, false otherwise.
+   */
+  override def lteqv(x: A, y: A): Boolean =
+    compare(x, y) <= 0
+
+  /**
+   * Returns true if `x` < `y`, false otherwise.
+   */
+  override def lt(x: A, y: A): Boolean =
+    compare(x, y) < 0
+
+  /**
+   * Returns true if `x` >= `y`, false otherwise.
+   */
+  override def gteqv(x: A, y: A): Boolean =
+    compare(x, y) >= 0
+
+  /**
+   * Returns true if `x` > `y`, false otherwise.
+   */
+  override def gt(x: A, y: A): Boolean =
+    compare(x, y) > 0
+
+  /**
+   * Returns a new `Order[A]` instance that first compares by the original
+   * `Order` instance and uses the provided `Order` instance to "break ties".
+   *
+   * That is, `x.whenEqual(y)` creates an `Order` that first orders by `x` and
+   * then (if two elements are equal) falls back to `y` for the comparison.
+   */
+  def whenEqual(o: Order[A]): Order[A] = new Order[A] {
+    def compare(x: A, y: A) = {
+      val c = self.compare(x, y)
+      if (c == 0) o.compare(x, y)
+      else c
+    }
+  }
+
+  /**
+   * Convert a `Order[A]` to a `scala.math.Ordering[A]`
+   * instance.
+   */
+  def toOrdering: Ordering[A] = new Ordering[A] {
+    def compare(x: A, y: A): Int = self.compare(x, y)
+  }
+}
+
+trait OrderFunctions {
+  def compare[@sp A](x: A, y: A)(implicit ev: Order[A]): Int =
+    ev.compare(x, y)
+
+  def eqv[@sp A](x: A, y: A)(implicit ev: Order[A]): Boolean =
+    ev.eqv(x, y)
+  def neqv[@sp A](x: A, y: A)(implicit ev: Order[A]): Boolean =
+    ev.neqv(x, y)
+  def gt[@sp A](x: A, y: A)(implicit ev: Order[A]): Boolean =
+    ev.gt(x, y)
+  def gteqv[@sp A](x: A, y: A)(implicit ev: Order[A]): Boolean =
+    ev.gteqv(x, y)
+  def lt[@sp A](x: A, y: A)(implicit ev: Order[A]): Boolean =
+    ev.lt(x, y)
+  def lteqv[@sp A](x: A, y: A)(implicit ev: Order[A]): Boolean =
+    ev.lteqv(x, y)
+
+  def min[@sp A](x: A, y: A)(implicit ev: Order[A]): A =
+    ev.min(x, y)
+  def max[@sp A](x: A, y: A)(implicit ev: Order[A]): A =
+    ev.max(x, y)
+}
+
+object Order extends OrderFunctions {
+
+  /**
+   * Convert an implicit `Order[A]` to an `Order[B]` using the given
+   * function `f`.
+   */
+  def by[@sp A, @sp B](f: A => B)(implicit ev: Order[B]): Order[A] =
+    ev.on(f)
+
+  /**
+   * Define an `Order[A]` using the given function `f`.
+   */
+  def from[@sp A](f: (A, A) => Int): Order[A] =
+    new Order[A] {
+      def compare(x: A, y: A) = f(x, y)
+    }
+
+  /**
+   * Implicitly convert a `Order[A]` to a `scala.math.Ordering[A]`
+   * instance.
+   */
+  implicit def ordering[A](implicit ev: Order[A]): Ordering[A] =
+    ev.toOrdering
+
+  /**
+   * An `Order` instance that considers all `A` instances to be equal.
+   */
+  def allEqual[A]: Order[A] = new Order[A] {
+    def compare(x: A, y: A): Int = 0
+  }
+
+
+  /**
+   * A `Monoid[Order[A]]` can be generated for all `A` with the following
+   * properties:
+   *
+   * `empty` returns a trivial `Order[A]` which considers all `A` instances to
+   * be equal.
+   *
+   * `combine(x: Order[A], y: Order[A])` creates an `Order[A]` that first
+   * orders by `x` and then (if two elements are equal) falls back to `y`.
+   *
+   * @see [[Order.whenEqual]]
+   */
+  def whenEqualMonoid[A]: Monoid[Order[A]] =
+    new Monoid[Order[A]] {
+      val empty: Order[A] = allEqual[A]
+
+      def combine(x: Order[A], y: Order[A]): Order[A] = x whenEqual y
+    }
+
+  def fromOrdering[A](implicit ev: Ordering[A]): Order[A] = new Order[A] {
+    def compare(x: A, y: A): Int = ev.compare(x, y)
+
+    override def toOrdering: Ordering[A] = ev
+  }
+}
diff --git a/kernel/src/main/scala/cats/PartialOrder.scala b/kernel/src/main/scala/cats/PartialOrder.scala
new file mode 100644
index 0000000000..a64eab6158
--- /dev/null
+++ b/kernel/src/main/scala/cats/PartialOrder.scala
@@ -0,0 +1,165 @@
+package cats
+
+import simulacrum.typeclass
+import scala.{specialized => sp}
+
+/**
+ * The `PartialOrder` type class is used to define a partial ordering on some type `A`.
+ * 
+ * A partial order is defined by a relation <=, which obeys the following laws:
+ * 
+ * - x <= x (reflexivity)
+ * - if x <= y and y <= x, then x = y (anti-symmetry)
+ * - if x <= y and y <= z, then x <= z (transitivity)
+ * 
+ * To compute both <= and >= at the same time, we use a Double number
+ * to encode the result of the comparisons x <= y and x >= y.
+ * The truth table is defined as follows:
+ * 
+ * x <= y    x >= y      Double
+ * true      true        = 0.0     (corresponds to x = y)
+ * false     false       = NaN     (x and y cannot be compared)
+ * true      false       = -1.0    (corresponds to x < y)
+ * false     true        = 1.0     (corresponds to x > y)
+ */
+@typeclass trait PartialOrder[@sp A] extends Eq[A] { self =>
+
+  /**
+   * Result of comparing `x` with `y`. Returns NaN if operands are not
+   * comparable. If operands are comparable, returns a Double whose
+   * sign is:
+   * - negative iff `x < y`
+   * - zero     iff `x = y`
+   * - positive iff `x > y`
+   */
+  def partialCompare(x: A, y: A): Double
+
+  /**
+   * Result of comparing `x` with `y`. Returns None if operands are
+   * not comparable. If operands are comparable, returns Some[Int]
+   * where the Int sign is:
+   * - negative iff `x < y`
+   * - zero     iff `x = y`
+   * - positive iff `x > y`
+   */  
+  def tryCompare(x: A, y: A): Option[Int] = {
+    val c = partialCompare(x, y)
+    if (c.isNaN) None else Some(c.signum)
+  }
+
+  /**
+   * Returns Some(x) if x <= y, Some(y) if x > y, otherwise None.
+   */
+  def pmin(x: A, y: A): Option[A] = {
+    val c = partialCompare(x, y)
+    if (c <= 0) Some(x)
+    else if (c > 0) Some(y)
+    else None
+  }
+
+  /**
+   * Returns Some(x) if x >= y, Some(y) if x < y, otherwise None.
+   */
+  def pmax(x: A, y: A): Option[A] = {
+    val c = partialCompare(x, y)
+    if (c >= 0)  Some(x)
+    else if (c < 0) Some(y)
+    else None
+  }
+
+  /**
+   * Defines a partial order on `B` by mapping `B` to `A` using `f`
+   * and using `A`s order to order `B`.
+   */
+  override def on[@sp B](f: B => A): PartialOrder[B] =
+    new PartialOrder[B] {
+      def partialCompare(x: B, y: B): Double = self.partialCompare(f(x), f(y))
+    }
+
+  /**
+   * Defines a partial order on `A` where all arrows switch direction.
+   */
+  def reverse: PartialOrder[A] =
+    new PartialOrder[A] {
+      def partialCompare(x: A, y: A): Double = self.partialCompare(y, x)
+    }
+
+  // The following may be overridden for performance:
+
+  /**
+   * Returns true if `x` = `y`, false otherwise.
+   */
+  override def eqv(x: A, y: A): Boolean = partialCompare(x, y) == 0
+
+  /**
+   * Returns true if `x` <= `y`, false otherwise.
+   */
+  def lteqv(x: A, y: A): Boolean = partialCompare(x, y) <= 0
+
+  /**
+   * Returns true if `x` < `y`, false otherwise.
+   */
+  def lt(x: A, y: A): Boolean = partialCompare(x, y) < 0
+
+  /**
+   * Returns true if `x` >= `y`, false otherwise.
+   */
+  def gteqv(x: A, y: A): Boolean = partialCompare(x, y) >= 0
+
+  /**
+   * Returns true if `x` > `y`, false otherwise.
+   */
+  def gt(x: A, y: A): Boolean = partialCompare(x, y) > 0
+}
+
+trait PartialOrderFunctions {
+
+  def partialCompare[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Double =
+    ev.partialCompare(x, y)
+  def tryCompare[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Option[Int] =
+    ev.tryCompare(x, y)
+
+  def pmin[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Option[A] =
+    ev.pmin(x, y)
+  def pmax[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Option[A] =
+    ev.pmax(x, y)
+
+  def eqv[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Boolean =
+    ev.eqv(x, y)
+  def lteqv[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Boolean =
+    ev.lteqv(x, y)
+  def lt[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Boolean =
+    ev.lt(x, y)
+  def gteqv[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Boolean =
+    ev.gteqv(x, y)
+  def gt[@sp A](x: A, y: A)(implicit ev: PartialOrder[A]): Boolean =
+    ev.gt(x, y)
+}
+
+object PartialOrder extends PartialOrderFunctions {
+
+  /**
+   * Convert an implicit `PartialOrder[A]` to an `PartialOrder[B]`
+   * using the given function `f`.
+   */
+  def by[@sp A, @sp B](f: A => B)(implicit ev: PartialOrder[B]): PartialOrder[A] =
+    ev.on(f)
+
+  /**
+   * Define a `PartialOrder[A]` using the given function `f`.
+   */
+  def from[@sp A](f: (A, A) => Double): PartialOrder[A] =
+    new PartialOrder[A] {
+      def partialCompare(x: A, y: A) = f(x, y)
+    }
+
+  /**
+   * Implicitly convert a `PartialOrder[A]` to a
+   * `scala.math.PartialOrdering[A]` instance.
+   */
+  implicit def partialOrdering[A](implicit ev: PartialOrder[A]): PartialOrdering[A] =
+    new PartialOrdering[A] {
+      def tryCompare(x: A, y: A): Option[Int] = ev.tryCompare(x, y)
+      def lteq(x: A, y: A): Boolean = ev.lteqv(x, y)
+    }
+}
diff --git a/kernel/src/main/scala/cats/Semigroup.scala b/kernel/src/main/scala/cats/Semigroup.scala
new file mode 100644
index 0000000000..ce0bed46a9
--- /dev/null
+++ b/kernel/src/main/scala/cats/Semigroup.scala
@@ -0,0 +1,68 @@
+package cats
+
+import simulacrum.typeclass
+import scala.{ specialized => sp }
+import scala.annotation.{ switch, tailrec }
+
+/**
+ * A semigroup is any set `A` with an associative operation (`combine`).
+ */
+@typeclass trait Semigroup[@sp(Int, Long, Float, Double) A] {
+
+  /**
+   * Associative operation taking which combines two values.
+   */
+  def combine(x: A, y: A): A
+
+  /**
+   * Return `a` combined with itself `n` times.
+   */
+  def combineN(a: A, n: Int): A =
+    if (n <= 0) throw new IllegalArgumentException("Repeated combining for semigroups must have n > 0")
+    else repeatedCombineN(a, n)
+
+  /**
+   * Return `a` combined with itself more than once.
+   */
+  protected[this] def repeatedCombineN(a: A, n: Int): A = {
+    @tailrec def loop(b: A, k: Int, extra: A): A =
+      if (k == 1) combine(b, extra) else {
+        val x = if ((k & 1) == 1) combine(b, extra) else extra
+        loop(combine(b, b), k >>> 1, x)
+      }
+    if (n == 1) a else loop(a, n - 1, a)
+  }
+
+  /**
+   * Given a sequence of `as`, combine them and return the total.
+   *
+   * If the sequence is empty, returns None. Otherwise, returns Some(total).
+   */
+  def combineAllOption(as: TraversableOnce[A]): Option[A] =
+    as.reduceOption(combine)
+}
+
+trait SemigroupFunctions[S[T] <: Semigroup[T]] {
+  def combine[@sp(Int, Long, Float, Double) A](x: A, y: A)(implicit ev: S[A]): A =
+    ev.combine(x, y)
+
+  def maybeCombine[@sp(Int, Long, Float, Double) A](ox: Option[A], y: A)(implicit ev: S[A]): A =
+    ox match {
+      case Some(x) => ev.combine(x, y)
+      case None => y
+    }
+
+  def maybeCombine[@sp(Int, Long, Float, Double) A](x: A, oy: Option[A])(implicit ev: S[A]): A =
+    oy match {
+      case Some(y) => ev.combine(x, y)
+      case None => x
+    }
+
+  def combineN[@sp(Int, Long, Float, Double) A](a: A, n: Int)(implicit ev: S[A]): A =
+    ev.combineN(a, n)
+
+  def combineAllOption[A](as: TraversableOnce[A])(implicit ev: S[A]): Option[A] =
+    ev.combineAllOption(as)
+}
+
+object Semigroup extends SemigroupFunctions[Semigroup]
diff --git a/laws/src/main/scala/cats/laws/GroupLaws.scala b/laws/src/main/scala/cats/laws/GroupLaws.scala
new file mode 100644
index 0000000000..05e24bf681
--- /dev/null
+++ b/laws/src/main/scala/cats/laws/GroupLaws.scala
@@ -0,0 +1,68 @@
+package cats
+package laws
+
+import org.typelevel.discipline.Laws
+
+import org.scalacheck.{Arbitrary, Prop}
+
+object GroupLaws {
+  def apply[A : Eq : Arbitrary] = new GroupLaws[A] {
+    def Equ = Eq[A]
+    def Arb = implicitly[Arbitrary[A]]
+  }
+}
+
+trait GroupLaws[A] extends Laws {
+
+  implicit def Equ: Eq[A]
+  implicit def Arb: Arbitrary[A]
+
+  // groups
+
+  def semigroup(implicit A: Semigroup[A]) = new GroupProperties(
+    name = "semigroup",
+    parents = Nil,
+    Rules.serializable(A),
+    Rules.associativity(A.combine),
+    Rules.repeat1("combineN")(A.combineN),
+    Rules.repeat2("combineN", "|+|")(A.combineN)(A.combine)
+  )
+
+  def monoid(implicit A: Monoid[A]) = new GroupProperties(
+    name = "monoid",
+    parents = List(semigroup),
+    Rules.leftIdentity(A.empty)(A.combine),
+    Rules.rightIdentity(A.empty)(A.combine),
+    Rules.repeat0("combineN", "id", A.empty)(A.combineN),
+    Rules.collect0("combineAll", "id", A.empty)(A.combineAll),
+    Rules.isId("isEmpty", A.empty)(A.isEmpty)
+  )
+
+  def group(implicit A: Group[A]) = new GroupProperties(
+    name = "group",
+    parents = List(monoid),
+    Rules.leftInverse(A.empty)(A.combine)(A.inverse),
+    Rules.rightInverse(A.empty)(A.combine)(A.inverse),
+    Rules.consistentInverse("remove")(A.remove)(A.combine)(A.inverse)
+  )
+
+  // property classes
+
+  class GroupProperties(
+    val name: String,
+    val parents: Seq[GroupProperties],
+    val props: (String, Prop)*
+  ) extends RuleSet {
+    val bases = Nil
+  }
+
+  class AdditiveProperties(
+    val base: GroupProperties,
+    val parents: Seq[AdditiveProperties],
+    val props: (String, Prop)*
+  ) extends RuleSet {
+    val name = base.name
+    val bases = List("base" -> base)
+  }
+
+}
diff --git a/laws/src/main/scala/cats/laws/OrderLaws.scala b/laws/src/main/scala/cats/laws/OrderLaws.scala
new file mode 100644
index 0000000000..8bb16b0dde
--- /dev/null
+++ b/laws/src/main/scala/cats/laws/OrderLaws.scala
@@ -0,0 +1,79 @@
+package cats
+package laws
+
+import org.typelevel.discipline.Laws
+
+import org.scalacheck.{Arbitrary, Prop}
+import org.scalacheck.Prop._
+
+import cats.std.boolean._
+
+object OrderLaws {
+  def apply[A: Eq: Arbitrary] = new OrderLaws[A] {
+    def Equ = Eq[A]
+    def Arb = implicitly[Arbitrary[A]]
+  }
+}
+
+trait OrderLaws[A] extends Laws {
+
+  implicit def Equ: Eq[A]
+  implicit def Arb: Arbitrary[A]
+
+  def eqv = new OrderProperties(
+    name = "eq",
+    parent = None,
+    Rules.serializable(Equ),
+    "reflexitivity-eq" -> forAll { (x: A) =>
+      x ?== x
+    },
+    "symmetry-eq" -> forAll { (x: A, y: A) =>
+      Equ.eqv(x, y) ?== Equ.eqv(y, x)
+    },
+    "antisymmetry-eq" -> forAll { (x: A, y: A, f: A => A) =>
+      !Equ.eqv(x, y) ?|| Equ.eqv(f(x), f(y))
+    },
+    "transitivity-eq" -> forAll { (x: A, y: A, z: A) =>
+      !(Equ.eqv(x, y) && Equ.eqv(y, z)) ?|| Equ.eqv(x, z)
+    }
+  )
+
+  def partialOrder(implicit A: PartialOrder[A]) = new OrderProperties(
+    name = "partialOrder",
+    parent = Some(eqv),
+    Rules.serializable(A),
+    "reflexitivity" -> forAll { (x: A) =>
+      x ?<= x
+    },
+    "antisymmetry" -> forAll { (x: A, y: A) =>
+      !(A.lteqv(x, y) && A.lteqv(y, x)) ?|| A.eqv(x, y)
+    },
+    "transitivity" -> forAll { (x: A, y: A, z: A) =>
+      !(A.lteqv(x, y) && A.lteqv(y, z)) ?|| A.lteqv(x, z)
+    },
+    "gteqv" -> forAll { (x: A, y: A) =>
+      A.lteqv(x, y) ?== A.gteqv(y, x)
+    },
+    "lt" -> forAll { (x: A, y: A) =>
+      A.lt(x, y) ?== (A.lteqv(x, y) && A.neqv(x, y))
+    },
+    "gt" -> forAll { (x: A, y: A) =>
+      A.lt(x, y) ?== A.gt(y, x)
+    }
+  )
+
+  def order(implicit A: Order[A]) = new OrderProperties(
+    name = "order",
+    parent = Some(partialOrder),
+    "totality" -> forAll { (x: A, y: A) =>
+      A.lteqv(x, y) ?|| A.lteqv(y, x)
+    }
+  )
+
+  class OrderProperties(
+    name: String,
+    parent: Option[RuleSet],
+    props: (String, Prop)*
+  ) extends DefaultRuleSet(name, parent, props: _*)
+
+}
diff --git a/laws/src/main/scala/cats/laws/Rules.scala b/laws/src/main/scala/cats/laws/Rules.scala
new file mode 100644
index 0000000000..885c52e53c
--- /dev/null
+++ b/laws/src/main/scala/cats/laws/Rules.scala
@@ -0,0 +1,98 @@
+package cats
+package laws
+
+import org.scalacheck.{Arbitrary, Prop}
+import org.scalacheck.Prop._
+import Prop.{False, Proof, Result}
+
+import cats.std.boolean._
+
+object Rules {
+
+  def serializable[M](m: M): (String, Prop) = "serializable" -> SerializableLaws.serializable(m)
+ 
+  // Comparison operators for testing are supplied by CheckEqOps and
+  // CheckOrderOps in package.scala. They are:
+  //
+  //    ?==  Ensure that x equals y
+  //    ?!=  Ensure that x does not equal y
+  //    ?<   Ensure that x < y
+  //    ?<=  Ensure that x <= y
+  //    ?>   Ensure that x > y
+  //    ?>=  Ensure that x >= y
+  //
+  // The reason to prefer these operators is that when tests fail, we
+  // will get more detaild output about what the failing values were
+  // (in addition to the input values generated by ScalaCheck).
+
+  def associativity[A: Arbitrary: Eq](f: (A, A) => A): (String, Prop) =
+    "associativity" -> forAll { (x: A, y: A, z: A) =>
+      f(f(x, y), z) ?== f(x, f(y, z))
+    }
+
+  def leftIdentity[A: Arbitrary: Eq](id: A)(f: (A, A) => A): (String, Prop) =
+    "leftIdentity" -> forAll { (x: A) =>
+      f(id, x) ?== x
+    }
+
+  def rightIdentity[A: Arbitrary: Eq](id: A)(f: (A, A) => A): (String, Prop) =
+    "rightIdentity" -> forAll { (x: A) =>
+      f(x, id) ?== x
+    }
+
+  def leftInverse[A: Arbitrary: Eq](id: A)(f: (A, A) => A)(inv: A => A): (String, Prop) =
+    "left inverse" -> forAll { (x: A) =>
+      id ?== f(inv(x), x)
+    }
+
+  def rightInverse[A: Arbitrary: Eq](id: A)(f: (A, A) => A)(inv: A => A): (String, Prop) =
+    "right inverse" -> forAll { (x: A) =>
+      id ?== f(x, inv(x))
+    }
+
+  def commutative[A: Arbitrary: Eq](f: (A, A) => A): (String, Prop) =
+    "commutative" -> forAll { (x: A, y: A) =>
+      f(x, y) ?== f(y, x)
+    }
+
+  def idempotence[A: Arbitrary: Eq](f: (A, A) => A): (String, Prop) =
+    "idempotence" -> forAll { (x: A) =>
+      f(x, x) ?== x
+    }
+
+  def consistentInverse[A: Arbitrary: Eq](name: String)(m: (A, A) => A)(f: (A, A) => A)(inv: A => A): (String, Prop) =
+    s"consistent $name" -> forAll { (x: A, y: A) =>
+      m(x, y) ?== f(x, inv(y))
+    }
+
+  def repeat0[A: Arbitrary: Eq](name: String, sym: String, id: A)(r: (A, Int) => A): (String, Prop) =
+    s"$name(a, 0) == $sym" -> forAll { (a: A) =>
+      r(a, 0) ?== id
+    }
+
+  def repeat1[A: Arbitrary: Eq](name: String)(r: (A, Int) => A): (String, Prop) =
+    s"$name(a, 1) == a" -> forAll { (a: A) =>
+      r(a, 1) ?== a
+    }
+
+  def repeat2[A: Arbitrary: Eq](name: String, sym: String)(r: (A, Int) => A)(f: (A, A) => A): (String, Prop) =
+    s"$name(a, 2) == a $sym a" -> forAll { (a: A) =>
+      r(a, 2) ?== f(a, a)
+    }
+
+  def collect0[A: Arbitrary: Eq](name: String, sym: String, id: A)(c: Seq[A] => A): (String, Prop) =
+    s"$name(Nil) == $sym" -> forAll { (a: A) =>
+      c(Nil) ?== id
+    }
+
+  def isId[A: Arbitrary: Eq](name: String, id: A)(p: A => Boolean): (String, Prop) =
+    name -> forAll { (x: A) =>
+      Eq.eqv(x, id) ?== p(x)
+    }
+
+  def distributive[A: Arbitrary: Eq](a: (A, A) => A)(m: (A, A) => A): (String, Prop) =
+    "distributive" → forAll { (x: A, y: A, z: A) =>
+      (m(x, a(y, z)) ?== a(m(x, y), m(x, z))) &&
+      (m(a(x, y), z) ?== a(m(x, z), m(y, z)))
+    }
+}
diff --git a/laws/src/main/scala/cats/laws/discipline/Eq.scala b/laws/src/main/scala/cats/laws/discipline/Eq.scala
index d79a1a6a84..ad4ba25219 100644
--- a/laws/src/main/scala/cats/laws/discipline/Eq.scala
+++ b/laws/src/main/scala/cats/laws/discipline/Eq.scala
@@ -2,7 +2,6 @@ package cats
 package laws
 package discipline
 
-import algebra.Eq
 import org.scalacheck.Arbitrary
 
 object eq {
diff --git a/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala b/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala
index 9c836c64f2..79e32eb712 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala
@@ -34,13 +34,12 @@ object MonoidalTests {
   }
 
   object Isomorphisms {
-    import algebra.laws._
     implicit def invariant[F[_]](implicit F: functor.Invariant[F]): Isomorphisms[F] =
       new Isomorphisms[F] {
         def associativity[A, B, C](fs: (F[(A, (B, C))], F[((A, B), C)]))(implicit EqFABC: Eq[F[(A, B, C)]]) =
-          F.imap(fs._1) { case (a, (b, c)) => (a, b, c) } { case (a, b, c) => (a, (b, c)) } ?==
+          F.imap(fs._1) { case (a, (b, c)) => (a, b, c) } { case (a, b, c) => (a, (b, c)) } <->
           F.imap(fs._2) { case ((a, b), c) => (a, b, c) } { case (a, b, c) => ((a, b), c) }
       }
   }
 
-}
\ No newline at end of file
+}
diff --git a/laws/src/main/scala/cats/laws/discipline/package.scala b/laws/src/main/scala/cats/laws/discipline/package.scala
index a015ad9bcd..de5f10dd6f 100644
--- a/laws/src/main/scala/cats/laws/discipline/package.scala
+++ b/laws/src/main/scala/cats/laws/discipline/package.scala
@@ -1,8 +1,6 @@
 package cats
 package laws
 
-import algebra.laws._
-
 import org.scalacheck.Prop
 
 package object discipline {
diff --git a/laws/src/main/scala/cats/laws/package.scala b/laws/src/main/scala/cats/laws/package.scala
index fbfcfaf7d7..5965ef27a1 100644
--- a/laws/src/main/scala/cats/laws/package.scala
+++ b/laws/src/main/scala/cats/laws/package.scala
@@ -1,7 +1,41 @@
 package cats
 
+import org.scalacheck._
+import org.scalacheck.util.Pretty
+import Prop.{False, Proof, Result}
+
 package object laws {
   implicit final class IsEqArrow[A](val lhs: A) extends AnyVal {
     def <->(rhs: A): IsEq[A] = IsEq(lhs, rhs)
   }
+
+  lazy val proved = Prop(Result(status = Proof))
+
+  lazy val falsified = Prop(Result(status = False))
+
+  object Ops {
+    def run[A](sym: String)(lhs: A, rhs: A)(f: (A, A) => Boolean): Prop =
+      if (f(lhs, rhs)) proved else falsified :| {
+        val exp = Pretty.pretty(lhs, Pretty.Params(0))
+        val got = Pretty.pretty(rhs, Pretty.Params(0))
+        s"($exp $sym $got) failed"
+      }
+  }
+
+  implicit class CheckEqOps[A](lhs: A)(implicit ev: Eq[A], pp: A => Pretty) {
+    def ?==(rhs: A): Prop = Ops.run("?==")(lhs, rhs)(ev.eqv)
+    def ?!=(rhs: A): Prop = Ops.run("?!=")(lhs, rhs)(ev.neqv)
+  }
+
+  implicit class CheckOrderOps[A](lhs: A)(implicit ev: PartialOrder[A], pp: A => Pretty) {
+    def ?<(rhs: A): Prop = Ops.run("?<")(lhs, rhs)(ev.lt)
+    def ?<=(rhs: A): Prop = Ops.run("?<=")(lhs, rhs)(ev.lteqv)
+    def ?>(rhs: A): Prop = Ops.run("?>")(lhs, rhs)(ev.gt)
+    def ?>=(rhs: A): Prop = Ops.run("?>=")(lhs, rhs)(ev.gteqv)
+  }
+
+  implicit class BooleanOps[A](lhs: Boolean)(implicit pp: Boolean => Pretty) {
+    def ?&&(rhs: Boolean): Prop = Ops.run("?&&")(lhs, rhs)(_ && _)
+    def ?||(rhs: Boolean): Prop = Ops.run("?||")(lhs, rhs)(_ || _)
+  }
 }
diff --git a/project/plugins.sbt b/project/plugins.sbt
index d5e4e7cb5c..06cde8b17e 100644
--- a/project/plugins.sbt
+++ b/project/plugins.sbt
@@ -10,3 +10,4 @@ addSbtPlugin("org.scoverage"       % "sbt-scoverage"          % "1.2.0")
 addSbtPlugin("com.typesafe.sbt"    % "sbt-git"                % "0.8.4")
 addSbtPlugin("org.scala-js"        % "sbt-scalajs"            % "0.6.5")
 addSbtPlugin("com.github.tkawachi" % "sbt-doctest"            % "0.3.5")
+addSbtPlugin("com.typesafe"        % "sbt-mima-plugin"        % "0.1.8")
diff --git a/tests/src/test/scala/cats/tests/ConstTests.scala b/tests/src/test/scala/cats/tests/ConstTests.scala
index c52482425a..73092ddf3f 100644
--- a/tests/src/test/scala/cats/tests/ConstTests.scala
+++ b/tests/src/test/scala/cats/tests/ConstTests.scala
@@ -1,10 +1,10 @@
 package cats
 package tests
 
-import algebra.laws.{GroupLaws, OrderLaws}
 
 import cats.data.{Const, NonEmptyList}
 import cats.functor.Contravariant
+import cats.laws.{GroupLaws, OrderLaws}
 import cats.laws.discipline._
 import cats.laws.discipline.arbitrary.{constArbitrary, oneAndArbitrary}
 
diff --git a/tests/src/test/scala/cats/tests/CoproductTests.scala b/tests/src/test/scala/cats/tests/CoproductTests.scala
index 7435cd3bfb..2f5ec19cf7 100644
--- a/tests/src/test/scala/cats/tests/CoproductTests.scala
+++ b/tests/src/test/scala/cats/tests/CoproductTests.scala
@@ -1,10 +1,9 @@
 package cats.tests
 
-import algebra.Eq
-import algebra.laws.OrderLaws
 import cats._
 import cats.data.Coproduct
 import cats.functor.Contravariant
+import cats.laws.OrderLaws
 import cats.laws.discipline._
 import cats.laws.discipline.arbitrary._
 import org.scalacheck.Arbitrary
diff --git a/tests/src/test/scala/cats/tests/EitherTests.scala b/tests/src/test/scala/cats/tests/EitherTests.scala
index 5c4a015968..c587edbffc 100644
--- a/tests/src/test/scala/cats/tests/EitherTests.scala
+++ b/tests/src/test/scala/cats/tests/EitherTests.scala
@@ -1,9 +1,9 @@
 package cats
 package tests
 
+import cats.laws.OrderLaws
 import cats.laws.discipline.{TraverseTests, MonadTests, SerializableTests, MonoidalTests}
 import cats.laws.discipline.eq._
-import algebra.laws.OrderLaws
 
 class EitherTests extends CatsSuite {
 
diff --git a/tests/src/test/scala/cats/tests/EvalTests.scala b/tests/src/test/scala/cats/tests/EvalTests.scala
index 7c40556908..28fe14a249 100644
--- a/tests/src/test/scala/cats/tests/EvalTests.scala
+++ b/tests/src/test/scala/cats/tests/EvalTests.scala
@@ -2,11 +2,10 @@ package cats
 package tests
 
 import scala.math.min
-import cats.laws.ComonadLaws
+import cats.laws.{ComonadLaws, GroupLaws, OrderLaws}
 import cats.laws.discipline.{MonoidalTests, BimonadTests, SerializableTests}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
-import algebra.laws.{GroupLaws, OrderLaws}
 
 class EvalTests extends CatsSuite {
 
diff --git a/tests/src/test/scala/cats/tests/FunctionTests.scala b/tests/src/test/scala/cats/tests/FunctionTests.scala
index 185dc8eaab..e81575e6a4 100644
--- a/tests/src/test/scala/cats/tests/FunctionTests.scala
+++ b/tests/src/test/scala/cats/tests/FunctionTests.scala
@@ -5,10 +5,10 @@ import org.scalacheck.Arbitrary
 
 import cats.arrow.{Arrow, Choice}
 import cats.functor.Contravariant
+import cats.laws.GroupLaws
 import cats.laws.discipline._
 import cats.laws.discipline.eq._
 import cats.laws.discipline.arbitrary._
-import algebra.laws.GroupLaws
 
 class FunctionTests extends CatsSuite {
 
diff --git a/tests/src/test/scala/cats/tests/KleisliTests.scala b/tests/src/test/scala/cats/tests/KleisliTests.scala
index 9c5f6cea22..9fc1fd8a03 100644
--- a/tests/src/test/scala/cats/tests/KleisliTests.scala
+++ b/tests/src/test/scala/cats/tests/KleisliTests.scala
@@ -4,11 +4,11 @@ package tests
 import cats.arrow.{Arrow, Choice, Split}
 import cats.data.{Kleisli, Reader}
 import cats.functor.{Contravariant, Strong}
+import cats.laws.GroupLaws
 import cats.laws.discipline._
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
 import org.scalacheck.Arbitrary
-import algebra.laws.GroupLaws
 import cats.laws.discipline.{SemigroupKTests, MonoidKTests}
 
 class KleisliTests extends CatsSuite {
diff --git a/tests/src/test/scala/cats/tests/OneAndTests.scala b/tests/src/test/scala/cats/tests/OneAndTests.scala
index 0b6f5c3ea0..30fdd4f863 100644
--- a/tests/src/test/scala/cats/tests/OneAndTests.scala
+++ b/tests/src/test/scala/cats/tests/OneAndTests.scala
@@ -1,9 +1,8 @@
 package cats
 package tests
 
-import algebra.laws.{GroupLaws, OrderLaws}
-
 import cats.data.{NonEmptyList, OneAnd}
+import cats.laws.{GroupLaws, OrderLaws}
 import cats.laws.discipline.{ComonadTests, FunctorTests, SemigroupKTests, FoldableTests, MonadTests, SerializableTests, MonoidalTests}
 import cats.laws.discipline.arbitrary.{evalArbitrary, oneAndArbitrary}
 import cats.laws.discipline.eq._
diff --git a/tests/src/test/scala/cats/tests/SetTests.scala b/tests/src/test/scala/cats/tests/SetTests.scala
index 571b238b0b..d32ce8d01b 100644
--- a/tests/src/test/scala/cats/tests/SetTests.scala
+++ b/tests/src/test/scala/cats/tests/SetTests.scala
@@ -4,7 +4,7 @@ package tests
 import cats.laws.discipline.{FoldableTests, MonoidKTests, SerializableTests}
 
 class SetTests extends CatsSuite {
-  checkAll("Set[Int]", algebra.laws.GroupLaws[Set[Int]].monoid)
+  checkAll("Set[Int]", cats.laws.GroupLaws[Set[Int]].monoid)
 
   checkAll("Set[Int]", MonoidKTests[Set].monoidK[Int])
   checkAll("MonoidK[Set]", SerializableTests.serializable(MonoidK[Set]))
diff --git a/tests/src/test/scala/cats/tests/StreamingTTests.scala b/tests/src/test/scala/cats/tests/StreamingTTests.scala
index 7140b9d496..6442ff3964 100644
--- a/tests/src/test/scala/cats/tests/StreamingTTests.scala
+++ b/tests/src/test/scala/cats/tests/StreamingTTests.scala
@@ -1,9 +1,8 @@
 package cats
 package tests
 
-import algebra.laws.OrderLaws
-
 import cats.data.{Streaming, StreamingT}
+import cats.laws.OrderLaws
 import cats.laws.discipline.{MonoidalTests, CoflatMapTests, MonadCombineTests, SerializableTests}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
diff --git a/tests/src/test/scala/cats/tests/StreamingTests.scala b/tests/src/test/scala/cats/tests/StreamingTests.scala
index 3097c5ce35..9dbb7568dc 100644
--- a/tests/src/test/scala/cats/tests/StreamingTests.scala
+++ b/tests/src/test/scala/cats/tests/StreamingTests.scala
@@ -1,9 +1,8 @@
 package cats
 package tests
 
-import algebra.laws.OrderLaws
-
 import cats.data.Streaming
+import cats.laws.OrderLaws
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
 import cats.laws.discipline.{TraverseTests, CoflatMapTests, MonadCombineTests, SerializableTests, MonoidalTests}
diff --git a/tests/src/test/scala/cats/tests/SyntaxTests.scala b/tests/src/test/scala/cats/tests/SyntaxTests.scala
index 098ed3e359..19fd969de1 100644
--- a/tests/src/test/scala/cats/tests/SyntaxTests.scala
+++ b/tests/src/test/scala/cats/tests/SyntaxTests.scala
@@ -3,8 +3,8 @@ package tests
 
 import cats.std.AllInstances
 import cats.syntax.AllSyntax
-import algebra.laws.GroupLaws
 import cats.functor.{Invariant, Contravariant}
+import cats.laws.GroupLaws
 import cats.laws.discipline.SerializableTests
 
 import org.scalacheck.{Arbitrary}
diff --git a/tests/src/test/scala/cats/tests/ValidatedTests.scala b/tests/src/test/scala/cats/tests/ValidatedTests.scala
index 683f774b1f..d3fb5b2906 100644
--- a/tests/src/test/scala/cats/tests/ValidatedTests.scala
+++ b/tests/src/test/scala/cats/tests/ValidatedTests.scala
@@ -6,9 +6,9 @@ import cats.data.Validated.{Valid, Invalid}
 import cats.laws.discipline.{BifunctorTests, TraverseTests, ApplicativeErrorTests, SerializableTests, MonoidalTests}
 import org.scalacheck.{Gen, Arbitrary}
 import org.scalacheck.Arbitrary._
+import cats.laws.{OrderLaws, GroupLaws}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq.tuple3Eq
-import algebra.laws.{OrderLaws, GroupLaws}
 
 import scala.util.Try
 
diff --git a/tests/src/test/scala/cats/tests/WriterTTests.scala b/tests/src/test/scala/cats/tests/WriterTTests.scala
index d214a08899..da1f13cdf5 100644
--- a/tests/src/test/scala/cats/tests/WriterTTests.scala
+++ b/tests/src/test/scala/cats/tests/WriterTTests.scala
@@ -3,11 +3,11 @@ package tests
 
 import cats.data.{Writer, WriterT}
 import cats.functor.Bifunctor
+import cats.laws.OrderLaws
 import cats.laws.discipline._
 import cats.laws.discipline.eq._
 import cats.laws.discipline.arbitrary._
 
-import algebra.laws.OrderLaws
 import org.scalacheck.Prop.forAll
 
 class WriterTTests extends CatsSuite {
diff --git a/tests/src/test/scala/cats/tests/XorTTests.scala b/tests/src/test/scala/cats/tests/XorTTests.scala
index 419036405f..b1e152df52 100644
--- a/tests/src/test/scala/cats/tests/XorTTests.scala
+++ b/tests/src/test/scala/cats/tests/XorTTests.scala
@@ -3,10 +3,10 @@ package tests
 
 import cats.functor.Bifunctor
 import cats.data.{Xor, XorT}
+import cats.laws.OrderLaws
 import cats.laws.discipline._
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq.tuple3Eq
-import algebra.laws.OrderLaws
 
 class XorTTests extends CatsSuite {
   implicit val eq0 = XorT.xorTEq[List, String, String Xor Int]
diff --git a/tests/src/test/scala/cats/tests/XorTests.scala b/tests/src/test/scala/cats/tests/XorTests.scala
index e8aec6c09b..8ac4d9d022 100644
--- a/tests/src/test/scala/cats/tests/XorTests.scala
+++ b/tests/src/test/scala/cats/tests/XorTests.scala
@@ -3,10 +3,10 @@ package tests
 
 import cats.data.{NonEmptyList, Xor, XorT}
 import cats.data.Xor._
+import cats.laws.{GroupLaws, OrderLaws}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.{BifunctorTests, TraverseTests, MonadErrorTests, SerializableTests, MonoidalTests}
 import cats.laws.discipline.eq.tuple3Eq
-import algebra.laws.{GroupLaws, OrderLaws}
 import org.scalacheck.{Arbitrary, Gen}
 import org.scalacheck.Arbitrary._