diff --git a/core/src/main/scala/cats/Applicative.scala b/core/src/main/scala/cats/Applicative.scala
index 70bdbf40d64..9acdb26e3cf 100644
--- a/core/src/main/scala/cats/Applicative.scala
+++ b/core/src/main/scala/cats/Applicative.scala
@@ -12,7 +12,7 @@ import simulacrum.typeclass
  *
  * Must obey the laws defined in cats.laws.ApplicativeLaws.
  */
-@typeclass trait Applicative[F[_]] extends Apply[F] { self =>
+@typeclass trait Applicative[F[_]] extends Apply[F] with Monoidal[F] { self =>
 
   /**
    * `pure` lifts any value into the Applicative Functor.
@@ -26,7 +26,7 @@ import simulacrum.typeclass
    *
    * This variant supports optional laziness.
    */
-  def pureEval[A](x: Eval[A]): F[A] = pure(x.value)
+  override def pureEval[A](x: Eval[A]): F[A] = pure(x.value)
 
   /**
    * Two sequentially dependent Applicatives can be composed.
diff --git a/core/src/main/scala/cats/Apply.scala b/core/src/main/scala/cats/Apply.scala
index 153bd68c271..68eacbe0222 100644
--- a/core/src/main/scala/cats/Apply.scala
+++ b/core/src/main/scala/cats/Apply.scala
@@ -8,7 +8,7 @@ import simulacrum.typeclass
  * Must obey the laws defined in cats.laws.ApplyLaws.
  */
 @typeclass(excludeParents=List("ApplyArityFunctions"))
-trait Apply[F[_]] extends Functor[F] with Monoidal[F] with ApplyArityFunctions[F] { self =>
+trait Apply[F[_]] extends Functor[F] with Semigroupal[F] with ApplyArityFunctions[F] { self =>
 
   /**
    * Given a value and a function in the Apply context, applies the
diff --git a/core/src/main/scala/cats/Monoidal.scala b/core/src/main/scala/cats/Monoidal.scala
index 3b56bce1502..f864265ac2f 100644
--- a/core/src/main/scala/cats/Monoidal.scala
+++ b/core/src/main/scala/cats/Monoidal.scala
@@ -8,8 +8,16 @@ import simulacrum.typeclass
  *
  * It is worth noting that the couple Monoidal and [[Functor]] is interdefinable with [[Apply]].
  */
-@typeclass trait Monoidal[F[_]] {
-  def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
-}
+@typeclass trait Monoidal[F[_]] extends Semigroupal[F] {
+  /**
+   * `pure` lifts any value into the lax monoidal functor.
+   */
+  def pure[A](x: A): F[A]
 
-object Monoidal extends MonoidalArityFunctions
\ No newline at end of file
+  /**
+   * `pureEval` lifts any value into the lax monoidal functor.
+   *
+   * This variant supports optional laziness.
+   */
+  def pureEval[A](x: Eval[A]): F[A] = pure(x.value)
+}
diff --git a/core/src/main/scala/cats/Semigroupal.scala b/core/src/main/scala/cats/Semigroupal.scala
new file mode 100644
index 00000000000..4d1c3f640e1
--- /dev/null
+++ b/core/src/main/scala/cats/Semigroupal.scala
@@ -0,0 +1,16 @@
+package cats
+
+import simulacrum.typeclass
+
+/**
+ * [[Semigroupal]] generalizes [[Functor]] by allowing functions of any
+ * arity to be applied to an effectful value. It encodes this by pairing
+ * together two effectful values (which can then be extended to be three
+ * or more) within a context, which can then be mapped over via the [[Functor]]
+ * instance.
+ */
+@typeclass trait Semigroupal[F[_]] extends Functor[F] {
+  def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
+}
+
+object Semigroupal extends SemigroupalArityFunctions
diff --git a/core/src/main/scala/cats/data/XorT.scala b/core/src/main/scala/cats/data/XorT.scala
index b50282cfe32..86e9587e1c1 100644
--- a/core/src/main/scala/cats/data/XorT.scala
+++ b/core/src/main/scala/cats/data/XorT.scala
@@ -120,7 +120,7 @@ final case class XorT[F[_], A, B](value: F[A Xor B]) {
    * {{{
    * scala> import cats.std.option._
    * scala> import cats.std.list._
-   * scala> import cats.syntax.monoidal._
+   * scala> import cats.syntax.semigroupal._
    * scala> type Error = String
    * scala> val v1: Validated[NonEmptyList[Error], Int] = Validated.Invalid(NonEmptyList("error 1"))
    * scala> val v2: Validated[NonEmptyList[Error], Int] = Validated.Invalid(NonEmptyList("error 2"))
diff --git a/core/src/main/scala/cats/syntax/all.scala b/core/src/main/scala/cats/syntax/all.scala
index dfe5cb667b3..fd6bb3f00bb 100644
--- a/core/src/main/scala/cats/syntax/all.scala
+++ b/core/src/main/scala/cats/syntax/all.scala
@@ -3,7 +3,6 @@ package syntax
 
 trait AllSyntax
     extends ApplySyntax
-    with MonoidalSyntax
     with BifunctorSyntax
     with CoflatMapSyntax
     with ComonadSyntax
@@ -25,6 +24,7 @@ trait AllSyntax
     with ProfunctorSyntax
     with ReducibleSyntax
     with SemigroupSyntax
+    with SemigroupalSyntax
     with SemigroupKSyntax
     with Show.ToShowOps
     with SplitSyntax
diff --git a/core/src/main/scala/cats/syntax/monoidal.scala b/core/src/main/scala/cats/syntax/monoidal.scala
index b4126200b53..8859c5d6d03 100644
--- a/core/src/main/scala/cats/syntax/monoidal.scala
+++ b/core/src/main/scala/cats/syntax/monoidal.scala
@@ -1,25 +1,25 @@
 package cats
 package syntax
 
-trait MonoidalSyntax1 {
-  implicit def monoidalSyntaxU[FA](fa: FA)(implicit U: Unapply[Monoidal, FA]): MonoidalOps[U.M, U.A] =
-    new MonoidalOps[U.M, U.A] {
+trait SemigroupalSyntax1 {
+  implicit def monoidalSyntaxU[FA](fa: FA)(implicit U: Unapply[Semigroupal, FA]): SemigroupalOps[U.M, U.A] =
+    new SemigroupalOps[U.M, U.A] {
       val self = U.subst(fa)
       val typeClassInstance = U.TC
     }
 }
 
-trait MonoidalSyntax extends MonoidalSyntax1 {
-  implicit def monoidalSyntax[F[_], A](fa: F[A])(implicit F: Monoidal[F]): MonoidalOps[F, A] =
-    new MonoidalOps[F, A] {
+trait SemigroupalSyntax extends SemigroupalSyntax1 {
+  implicit def monoidalSyntax[F[_], A](fa: F[A])(implicit F: Semigroupal[F]): SemigroupalOps[F, A] =
+    new SemigroupalOps[F, A] {
       val self = fa
       val typeClassInstance = F
     }
 }
 
-abstract class MonoidalOps[F[_], A] extends Monoidal.Ops[F, A] {
-  def |@|[B](fb: F[B]): MonoidalBuilder[F]#MonoidalBuilder2[A, B] =
-    new MonoidalBuilder[F] |@| self |@| fb
+abstract class SemigroupalOps[F[_], A] extends Semigroupal.Ops[F, A] {
+  def |@|[B](fb: F[B]): SemigroupalBuilder[F]#SemigroupalBuilder2[A, B] =
+    new SemigroupalBuilder[F] |@| self |@| fb
 
   def *>[B](fb: F[B])(implicit F: Functor[F]): F[B] = F.map(typeClassInstance.product(self, fb)) { case (a, b) => b }
 
diff --git a/core/src/main/scala/cats/syntax/package.scala b/core/src/main/scala/cats/syntax/package.scala
index c61f8df9fe7..6b48f3dd124 100644
--- a/core/src/main/scala/cats/syntax/package.scala
+++ b/core/src/main/scala/cats/syntax/package.scala
@@ -3,7 +3,6 @@ package cats
 package object syntax {
   object all extends AllSyntax
   object apply extends ApplySyntax
-  object monoidal extends MonoidalSyntax
   object bifunctor extends BifunctorSyntax
   object coflatMap extends CoflatMapSyntax
   object comonad extends ComonadSyntax
@@ -23,6 +22,7 @@ package object syntax {
   object partialOrder extends PartialOrderSyntax
   object profunctor extends ProfunctorSyntax
   object semigroup extends SemigroupSyntax
+  object semigroupal extends SemigroupalSyntax
   object semigroupk extends SemigroupKSyntax
   object show extends Show.ToShowOps
   object split extends SplitSyntax
diff --git a/docs/src/main/tut/apply.md b/docs/src/main/tut/apply.md
index 6607ab928e6..c3c27c42497 100644
--- a/docs/src/main/tut/apply.md
+++ b/docs/src/main/tut/apply.md
@@ -125,7 +125,7 @@ In order to use it, first import `cats.syntax.all._` or `cats.syntax.apply._`.
 Here we see that the following two functions, `f1` and `f2`, are equivalent:
 
 ```tut
-import cats.syntax.monoidal._
+import cats.syntax.semigroupal._
 
 def f1(a: Option[Int], b: Option[Int], c: Option[Int]) =
   (a |@| b |@| c) map { _ * _ * _ }
diff --git a/docs/src/main/tut/freeapplicative.md b/docs/src/main/tut/freeapplicative.md
index 57a48fb14d8..613c791b68a 100644
--- a/docs/src/main/tut/freeapplicative.md
+++ b/docs/src/main/tut/freeapplicative.md
@@ -39,7 +39,7 @@ Because a `FreeApplicative` only supports the operations of `Applicative`, we do
 of a for-comprehension. We can however still use `Applicative` syntax provided by Cats.
 
 ```tut:silent
-import cats.syntax.monoidal._
+import cats.syntax.semigroupal._
 
 val prog: Validation[Boolean] = (size(5) |@| hasNumber).map { case (l, r) => l && r}
 ```
diff --git a/laws/src/main/scala/cats/laws/ApplicativeLaws.scala b/laws/src/main/scala/cats/laws/ApplicativeLaws.scala
index e963d17f634..1ce45fb2dc7 100644
--- a/laws/src/main/scala/cats/laws/ApplicativeLaws.scala
+++ b/laws/src/main/scala/cats/laws/ApplicativeLaws.scala
@@ -7,7 +7,7 @@ import cats.syntax.functor._
 /**
  * Laws that must be obeyed by any `Applicative`.
  */
-trait ApplicativeLaws[F[_]] extends ApplyLaws[F] {
+trait ApplicativeLaws[F[_]] extends ApplyLaws[F] with MonoidalLaws[F] {
   implicit override def F: Applicative[F]
 
   def applicativeIdentity[A](fa: F[A]): IsEq[F[A]] =
diff --git a/laws/src/main/scala/cats/laws/ApplyLaws.scala b/laws/src/main/scala/cats/laws/ApplyLaws.scala
index c6c2a768598..db2919b2fbf 100644
--- a/laws/src/main/scala/cats/laws/ApplyLaws.scala
+++ b/laws/src/main/scala/cats/laws/ApplyLaws.scala
@@ -7,7 +7,7 @@ import cats.syntax.functor._
 /**
  * Laws that must be obeyed by any `Apply`.
  */
-trait ApplyLaws[F[_]] extends FunctorLaws[F] with MonoidalLaws[F] {
+trait ApplyLaws[F[_]] extends FunctorLaws[F] with SemigroupalLaws[F] {
   implicit override def F: Apply[F]
 
   def applyComposition[A, B, C](fa: F[A], fab: F[A => B], fbc: F[B => C]): IsEq[F[C]] = {
diff --git a/laws/src/main/scala/cats/laws/MonoidalLaws.scala b/laws/src/main/scala/cats/laws/MonoidalLaws.scala
index 402e1cd9bf1..67acdd77361 100644
--- a/laws/src/main/scala/cats/laws/MonoidalLaws.scala
+++ b/laws/src/main/scala/cats/laws/MonoidalLaws.scala
@@ -1,13 +1,15 @@
 package cats
 package laws
 
-trait MonoidalLaws[F[_]] {
+trait MonoidalLaws[F[_]] extends SemigroupalLaws[F] {
 
   implicit def F: Monoidal[F]
 
-  def monoidalAssociativity[A, B, C](fa: F[A], fb: F[B], fc: F[C]): (F[(A, (B, C))], F[((A, B), C)]) =
-    (F.product(fa, F.product(fb, fc)), F.product(F.product(fa, fb), fc))
+  def leftIdentity[A](fa: F[A]): IsEq[F[A]] =
+    F.map(F.product(F.pure(()), fa))(_._2) <-> fa
 
+  def rightIdentity[A](fa: F[A]): IsEq[F[A]] =
+    F.map(F.product(fa, F.pure(())))(_._1) <-> fa
 }
 
 object MonoidalLaws {
@@ -15,4 +17,4 @@ object MonoidalLaws {
   def apply[F[_]](implicit ev: Monoidal[F]): MonoidalLaws[F] =
     new MonoidalLaws[F] { val F = ev }
 
-}
\ No newline at end of file
+}
diff --git a/laws/src/main/scala/cats/laws/SemigroupalLaws.scala b/laws/src/main/scala/cats/laws/SemigroupalLaws.scala
new file mode 100644
index 00000000000..7b73fad5e01
--- /dev/null
+++ b/laws/src/main/scala/cats/laws/SemigroupalLaws.scala
@@ -0,0 +1,18 @@
+package cats
+package laws
+
+trait SemigroupalLaws[F[_]] {
+
+  implicit def F: Semigroupal[F]
+
+  def semigroupalAssociativity[A, B, C](fa: F[A], fb: F[B], fc: F[C]): (F[(A, (B, C))], F[((A, B), C)]) =
+    (F.product(fa, F.product(fb, fc)), F.product(F.product(fa, fb), fc))
+
+}
+
+object SemigroupalLaws {
+
+  def apply[F[_]](implicit ev: Semigroupal[F]): SemigroupalLaws[F] =
+    new SemigroupalLaws[F] { val F = ev }
+
+}
diff --git a/laws/src/main/scala/cats/laws/discipline/AlternativeTests.scala b/laws/src/main/scala/cats/laws/discipline/AlternativeTests.scala
index e2f35ab31c9..b93173197f1 100644
--- a/laws/src/main/scala/cats/laws/discipline/AlternativeTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/AlternativeTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
diff --git a/laws/src/main/scala/cats/laws/discipline/ApplicativeTests.scala b/laws/src/main/scala/cats/laws/discipline/ApplicativeTests.scala
index 49117c8dea9..c2c3090f014 100644
--- a/laws/src/main/scala/cats/laws/discipline/ApplicativeTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/ApplicativeTests.scala
@@ -2,12 +2,12 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
 
-trait ApplicativeTests[F[_]] extends ApplyTests[F] {
+trait ApplicativeTests[F[_]] extends ApplyTests[F] with MonoidalTests[F] {
   def laws: ApplicativeLaws[F]
 
   def applicative[A: Arbitrary, B: Arbitrary, C: Arbitrary](implicit
@@ -22,13 +22,17 @@ trait ApplicativeTests[F[_]] extends ApplyTests[F] {
     EqFABC: Eq[F[(A, B, C)]],
     iso: Isomorphisms[F]
   ): RuleSet = {
-    new DefaultRuleSet(
-      name = "applicative",
-      parent = Some(apply[A, B, C]),
-      "applicative identity" -> forAll(laws.applicativeIdentity[A] _),
-      "applicative homomorphism" -> forAll(laws.applicativeHomomorphism[A, B] _),
-      "applicative interchange" -> forAll(laws.applicativeInterchange[A, B] _),
-      "applicative map" -> forAll(laws.applicativeMap[A, B] _))
+    new RuleSet {
+      def name: String = "applicative"
+      def bases: Seq[(String, RuleSet)] = Nil
+      def parents: Seq[RuleSet] = Seq(apply[A, B, C], monoidal[A, B, C])
+      def props: Seq[(String, Prop)] = Seq(
+        "applicative identity" -> forAll(laws.applicativeIdentity[A] _),
+        "applicative homomorphism" -> forAll(laws.applicativeHomomorphism[A, B] _),
+        "applicative interchange" -> forAll(laws.applicativeInterchange[A, B] _),
+        "applicative map" -> forAll(laws.applicativeMap[A, B] _)
+      )
+    }
   }
 }
 
diff --git a/laws/src/main/scala/cats/laws/discipline/ApplyTests.scala b/laws/src/main/scala/cats/laws/discipline/ApplyTests.scala
index d6ca9f88337..a179986aca3 100644
--- a/laws/src/main/scala/cats/laws/discipline/ApplyTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/ApplyTests.scala
@@ -2,12 +2,12 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
 
-trait ApplyTests[F[_]] extends FunctorTests[F] with MonoidalTests[F] {
+trait ApplyTests[F[_]] extends FunctorTests[F] with SemigroupalTests[F] {
   def laws: ApplyLaws[F]
 
   def apply[A: Arbitrary, B: Arbitrary, C: Arbitrary](implicit
@@ -22,7 +22,7 @@ trait ApplyTests[F[_]] extends FunctorTests[F] with MonoidalTests[F] {
     iso: Isomorphisms[F]
   ): RuleSet = new RuleSet {
     val name = "apply"
-    val parents = Seq(functor[A, B, C], monoidal[A, B, C])
+    val parents = Seq(functor[A, B, C], semigroupal[A, B, C])
     val bases = Seq.empty
     val props = Seq("apply composition" -> forAll(laws.applyComposition[A, B, C] _))
   }
diff --git a/laws/src/main/scala/cats/laws/discipline/BimonadTests.scala b/laws/src/main/scala/cats/laws/discipline/BimonadTests.scala
index a5a7fee0473..04854e96b29 100644
--- a/laws/src/main/scala/cats/laws/discipline/BimonadTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/BimonadTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
diff --git a/laws/src/main/scala/cats/laws/discipline/FlatMapTests.scala b/laws/src/main/scala/cats/laws/discipline/FlatMapTests.scala
index c4aebf3a56c..d013777da8d 100644
--- a/laws/src/main/scala/cats/laws/discipline/FlatMapTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/FlatMapTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
diff --git a/laws/src/main/scala/cats/laws/discipline/MonadCombineTests.scala b/laws/src/main/scala/cats/laws/discipline/MonadCombineTests.scala
index 7f3cbea2ad2..2d1b6e91107 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonadCombineTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonadCombineTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
diff --git a/laws/src/main/scala/cats/laws/discipline/MonadErrorTests.scala b/laws/src/main/scala/cats/laws/discipline/MonadErrorTests.scala
index 2b7ee10dfb5..1d09ed6b924 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonadErrorTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonadErrorTests.scala
@@ -3,7 +3,7 @@ package laws
 package discipline
 
 import cats.data.{ Xor, XorT }
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq.unitEq
 import org.scalacheck.{Arbitrary, Prop}
diff --git a/laws/src/main/scala/cats/laws/discipline/MonadFilterTests.scala b/laws/src/main/scala/cats/laws/discipline/MonadFilterTests.scala
index e56b7fe427e..df229d4e623 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonadFilterTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonadFilterTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
diff --git a/laws/src/main/scala/cats/laws/discipline/MonadReaderTests.scala b/laws/src/main/scala/cats/laws/discipline/MonadReaderTests.scala
index e3fcdd0fc24..8ca248fb0e5 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonadReaderTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonadReaderTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.{Arbitrary, Prop}
 import org.scalacheck.Prop.forAll
 
diff --git a/laws/src/main/scala/cats/laws/discipline/MonadStateTests.scala b/laws/src/main/scala/cats/laws/discipline/MonadStateTests.scala
index c9d5015c089..a85f8578504 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonadStateTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonadStateTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import eq.unitEq
 import org.scalacheck.{Arbitrary, Prop}
 import org.scalacheck.Prop.forAll
diff --git a/laws/src/main/scala/cats/laws/discipline/MonadTests.scala b/laws/src/main/scala/cats/laws/discipline/MonadTests.scala
index 5c676bec78c..15624b5b801 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonadTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonadTests.scala
@@ -2,7 +2,7 @@ package cats
 package laws
 package discipline
 
-import cats.laws.discipline.MonoidalTests.Isomorphisms
+import cats.laws.discipline.SemigroupalTests.Isomorphisms
 import org.scalacheck.Arbitrary
 import org.scalacheck.Prop
 import Prop._
diff --git a/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala b/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala
index 9c836c64f22..c39b7522fdc 100644
--- a/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/MonoidalTests.scala
@@ -7,40 +7,27 @@ import org.scalacheck.Prop
 import Prop._
 import org.typelevel.discipline.Laws
 
-trait MonoidalTests[F[_]] extends Laws {
+trait MonoidalTests[F[_]] extends SemigroupalTests[F] {
   def laws: MonoidalLaws[F]
 
   def monoidal[A : Arbitrary, B : Arbitrary, C : Arbitrary](implicit
-    iso: MonoidalTests.Isomorphisms[F],
+    iso: SemigroupalTests.Isomorphisms[F],
     ArbFA: Arbitrary[F[A]],
     ArbFB: Arbitrary[F[B]],
     ArbFC: Arbitrary[F[C]],
+    EqFA: Eq[F[A]],
     EqFABC: Eq[F[(A, B, C)]]
   ): RuleSet = {
     new DefaultRuleSet(
       name = "monoidal",
-      parent = None,
-      "monoidal associativity" -> forAll((fa: F[A], fb: F[B], fc: F[C]) => iso.associativity(laws.monoidalAssociativity(fa, fb, fc)))
+      parent = Some(semigroupal[A, B, C]),
+      "monoidal left identity" -> forAll(laws.leftIdentity[A] _),
+      "monoidal right identity" -> forAll(laws.rightIdentity[A] _)
     )
   }
 }
 
 object MonoidalTests {
-  def apply[F[_] : Monoidal](implicit ev: Isomorphisms[F]): MonoidalTests[F] =
+  def apply[F[_] : Monoidal](implicit ev: SemigroupalTests.Isomorphisms[F]): MonoidalTests[F] =
     new MonoidalTests[F] { val laws: MonoidalLaws[F] = MonoidalLaws[F] }
-
-  trait Isomorphisms[F[_]] {
-    def associativity[A, B, C](fs: (F[(A, (B, C))], F[((A, B), C)]))(implicit EqFABC: Eq[F[(A, B, C)]]): Prop
-  }
-
-  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._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/SemigroupalTests.scala b/laws/src/main/scala/cats/laws/discipline/SemigroupalTests.scala
new file mode 100644
index 00000000000..9091d6efbc0
--- /dev/null
+++ b/laws/src/main/scala/cats/laws/discipline/SemigroupalTests.scala
@@ -0,0 +1,46 @@
+package cats
+package laws
+package discipline
+
+import org.scalacheck.Arbitrary
+import org.scalacheck.Prop
+import Prop._
+import org.typelevel.discipline.Laws
+
+trait SemigroupalTests[F[_]] extends Laws {
+  def laws: SemigroupalLaws[F]
+
+  def semigroupal[A : Arbitrary, B : Arbitrary, C : Arbitrary](implicit
+    iso: SemigroupalTests.Isomorphisms[F],
+    ArbFA: Arbitrary[F[A]],
+    ArbFB: Arbitrary[F[B]],
+    ArbFC: Arbitrary[F[C]],
+    EqFABC: Eq[F[(A, B, C)]]
+  ): RuleSet = {
+    new DefaultRuleSet(
+      name = "semigroupal",
+      parent = None,
+      "semigroupal associativity" -> forAll((fa: F[A], fb: F[B], fc: F[C]) => iso.associativity(laws.semigroupalAssociativity(fa, fb, fc)))
+    )
+  }
+}
+
+object SemigroupalTests {
+  def apply[F[_] : Semigroupal](implicit ev: Isomorphisms[F]): SemigroupalTests[F] =
+    new SemigroupalTests[F] { val laws: SemigroupalLaws[F] = SemigroupalLaws[F] }
+
+  trait Isomorphisms[F[_]] {
+    def associativity[A, B, C](fs: (F[(A, (B, C))], F[((A, B), C)]))(implicit EqFABC: Eq[F[(A, B, C)]]): Prop
+  }
+
+  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._2) { case ((a, b), c) => (a, b, c) } { case (a, b, c) => ((a, b), c) }
+      }
+  }
+
+}
diff --git a/project/Boilerplate.scala b/project/Boilerplate.scala
index 9c542460c9b..a2a7bf60454 100644
--- a/project/Boilerplate.scala
+++ b/project/Boilerplate.scala
@@ -25,8 +25,8 @@ object Boilerplate {
 
 
   val templates: Seq[Template] = Seq(
-    GenMonoidalBuilders,
-    GenMonoidalArityFunctions,
+    GenSemigroupalBuilders,
+    GenSemigroupalArityFunctions,
     GenApplyArityFunctions
   )
 
@@ -85,8 +85,8 @@ object Boilerplate {
     The block otherwise behaves as a standard interpolated string with regards to variable substitution.
   */
 
-  object GenMonoidalBuilders extends Template {
-    def filename(root: File) = root /  "cats" / "syntax" / "MonoidalBuilder.scala"
+  object GenSemigroupalBuilders extends Template {
+    def filename(root: File) = root /  "cats" / "syntax" / "SemigroupalBuilder.scala"
 
     def content(tv: TemplateVals) = {
       import tv._
@@ -95,7 +95,7 @@ object Boilerplate {
       val tpesString = synTypes mkString ", "
       val params = (synVals zip tpes) map { case (v,t) => s"$v:$t"} mkString ", "
       val next = if (arity + 1 <= maxArity) {
-        s"def |@|[Z](z: F[Z]) = new MonoidalBuilder${arity + 1}(${`a..n`}, z)"
+        s"def |@|[Z](z: F[Z]) = new SemigroupalBuilder${arity + 1}(${`a..n`}, z)"
       } else {
         ""
       }
@@ -104,18 +104,18 @@ object Boilerplate {
 
       val map =
         if (arity == 1) s"def map[Z](f: (${`A..N`}) => Z)(implicit functor: Functor[F]): F[Z] = functor.map(${`a..n`})(f)"
-        else s"def map[Z](f: (${`A..N`}) => Z)(implicit functor: Functor[F], monoidal: Monoidal[F]): F[Z] = Monoidal.map$n(${`a..n`})(f)"
+        else s"def map[Z](f: (${`A..N`}) => Z)(implicit functor: Functor[F], semigroupal: Semigroupal[F]): F[Z] = Semigroupal.map$n(${`a..n`})(f)"
 
       val contramap =
         if (arity == 1) s"def contramap[Z](f: Z => (${`A..N`}))(implicit contravariant: Contravariant[F]): F[Z] = contravariant.contramap(${`a..n`})(f)"
-        else s"def contramap[Z](f: Z => (${`A..N`}))(implicit contravariant: Contravariant[F], monoidal: Monoidal[F]): F[Z] = Monoidal.contramap$n(${`a..n`})(f)"
+        else s"def contramap[Z](f: Z => (${`A..N`}))(implicit contravariant: Contravariant[F], semigroupal: Semigroupal[F]): F[Z] = Semigroupal.contramap$n(${`a..n`})(f)"
 
       val imap =
         if (arity == 1) s"def imap[Z](f: (${`A..N`}) => Z)(g: Z => (${`A..N`}))(implicit invariant: Invariant[F]): F[Z] = invariant.imap(${`a..n`})(f)(g)"
-        else s"def imap[Z](f: (${`A..N`}) => Z)(g: Z => (${`A..N`}))(implicit invariant: Invariant[F], monoidal: Monoidal[F]): F[Z] = Monoidal.imap$n(${`a..n`})(f)(g)"
+        else s"def imap[Z](f: (${`A..N`}) => Z)(g: Z => (${`A..N`}))(implicit invariant: Invariant[F], semigroupal: Semigroupal[F]): F[Z] = Semigroupal.imap$n(${`a..n`})(f)(g)"
 
       val tupled = if (arity != 1) {
-        s"def tupled(implicit invariant: Invariant[F], monoidal: Monoidal[F]): F[(${`A..N`})] = Monoidal.tuple$n(${`a..n`})"
+        s"def tupled(implicit invariant: Invariant[F], semigroupal: Semigroupal[F]): F[(${`A..N`})] = Semigroupal.tuple$n(${`a..n`})"
       } else {
         ""
       }
@@ -126,10 +126,10 @@ object Boilerplate {
         |
         |import cats.functor.{Contravariant, Invariant}
         |
-        |private[syntax] final class MonoidalBuilder[F[_]] {
-        |  def |@|[A](a: F[A]) = new MonoidalBuilder1(a)
+        |private[syntax] final class SemigroupalBuilder[F[_]] {
+        |  def |@|[A](a: F[A]) = new SemigroupalBuilder1(a)
         |
-        -  private[syntax] final class MonoidalBuilder$arity[${`A..N`}]($params) {
+        -  private[syntax] final class SemigroupalBuilder$arity[${`A..N`}]($params) {
         -    $next
         -    def ap[Z](f: F[(${`A..N`}) => Z])(implicit apply: Apply[F]): F[Z] = apply.ap$n(${`a..n`})(f)
         -    $map
@@ -172,17 +172,17 @@ object Boilerplate {
       block"""
         |package cats
         |trait ApplyArityFunctions[F[_]] { self: Apply[F] =>
-        |  def tuple2[A, B](f1: F[A], f2: F[B]): F[(A, B)] = Monoidal.tuple2(f1, f2)(self, self)
+        |  def tuple2[A, B](f1: F[A], f2: F[B]): F[(A, B)] = Semigroupal.tuple2(f1, f2)(self, self)
         -  def ap$arity[${`A..N`}, Z]($fparams)(f: F[(${`A..N`}) => Z]):F[Z] = $apply
-        -  def map$arity[${`A..N`}, Z]($fparams)(f: (${`A..N`}) => Z): F[Z] = Monoidal.map$arity($fparams)(f)(self, self)
-        -  def tuple$arity[${`A..N`}, Z]($fparams): F[(${`A..N`})] = Monoidal.tuple$arity($fparams)(self, self)
+        -  def map$arity[${`A..N`}, Z]($fparams)(f: (${`A..N`}) => Z): F[Z] = Semigroupal.map$arity($fparams)(f)(self, self)
+        -  def tuple$arity[${`A..N`}, Z]($fparams): F[(${`A..N`})] = Semigroupal.tuple$arity($fparams)(self, self)
         |}
       """
     }
   }
 
-  object GenMonoidalArityFunctions extends Template {
-    def filename(root: File) = root / "cats" / "MonoidalArityFunctions.scala"
+  object GenSemigroupalArityFunctions extends Template {
+    def filename(root: File) = root / "cats" / "SemigroupalArityFunctions.scala"
     override def range = 2 to maxArity
     def content(tv: TemplateVals) = {
       import tv._
@@ -192,19 +192,19 @@ object Boilerplate {
       val fparams = (fargs zip tpes) map { case (v,t) => s"$v:$t"} mkString ", "
       val fargsS = fargs mkString ", "
 
-      val nestedProducts = (0 until (arity - 2)).foldRight(s"monoidal.product(f${arity - 2}, f${arity - 1})")((i, acc) => s"monoidal.product(f$i, $acc)")
+      val nestedProducts = (0 until (arity - 2)).foldRight(s"semigroupal.product(f${arity - 2}, f${arity - 1})")((i, acc) => s"semigroupal.product(f$i, $acc)")
       val `nested (a..n)` = (0 until (arity - 2)).foldRight(s"(a${arity - 2}, a${arity - 1})")((i, acc) => s"(a$i, $acc)")
 
       block"""
          |package cats
-         |trait MonoidalArityFunctions {
-        -  def map$arity[F[_], ${`A..N`}, Z]($fparams)(f: (${`A..N`}) => Z)(implicit monoidal: Monoidal[F], functor: Functor[F]): F[Z] =
+         |trait SemigroupalArityFunctions {
+        -  def map$arity[F[_], ${`A..N`}, Z]($fparams)(f: (${`A..N`}) => Z)(implicit semigroupal: Semigroupal[F], functor: Functor[F]): F[Z] =
         -    functor.map($nestedProducts) { case ${`nested (a..n)`} => f(${`a..n`}) }
-        -  def contramap$arity[F[_], ${`A..N`}, Z]($fparams)(f: Z => (${`A..N`}))(implicit monoidal: Monoidal[F], contravariant: functor.Contravariant[F]):F[Z] =
+        -  def contramap$arity[F[_], ${`A..N`}, Z]($fparams)(f: Z => (${`A..N`}))(implicit semigroupal: Semigroupal[F], contravariant: functor.Contravariant[F]):F[Z] =
         -    contravariant.contramap($nestedProducts) { z => val ${`(a..n)`} = f(z); ${`nested (a..n)`} }
-        -  def imap$arity[F[_], ${`A..N`}, Z]($fparams)(f: (${`A..N`}) => Z)(g: Z => (${`A..N`}))(implicit monoidal: Monoidal[F], invariant: functor.Invariant[F]):F[Z] =
+        -  def imap$arity[F[_], ${`A..N`}, Z]($fparams)(f: (${`A..N`}) => Z)(g: Z => (${`A..N`}))(implicit semigroupal: Semigroupal[F], invariant: functor.Invariant[F]):F[Z] =
         -    invariant.imap($nestedProducts) { case ${`nested (a..n)`} => f(${`a..n`}) } { z => val ${`(a..n)`} = g(z); ${`nested (a..n)`} }
-        -  def tuple$arity[F[_], ${`A..N`}]($fparams)(implicit monoidal: Monoidal[F], invariant: functor.Invariant[F]):F[(${`A..N`})] =
+        -  def tuple$arity[F[_], ${`A..N`}]($fparams)(implicit semigroupal: Semigroupal[F], invariant: functor.Invariant[F]):F[(${`A..N`})] =
         -    imap$arity($fargsS)((${`_.._`}))(identity)
          |}
       """
diff --git a/tests/src/test/scala/cats/tests/CokleisliTests.scala b/tests/src/test/scala/cats/tests/CokleisliTests.scala
index a635e9b13e0..e9f7ce65527 100644
--- a/tests/src/test/scala/cats/tests/CokleisliTests.scala
+++ b/tests/src/test/scala/cats/tests/CokleisliTests.scala
@@ -18,10 +18,10 @@ class CokleisliTests extends SlowCatsSuite {
   def cokleisliEqE[F[_], A](implicit A: Arbitrary[F[A]], FA: Eq[A]): Eq[Cokleisli[F, A, A]] =
     Eq.by[Cokleisli[F, A, A], F[A] => A](_.run)
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Cokleisli[Option, Int, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Cokleisli[Option, Int, ?]]
 
-  checkAll("Cokleisli[Option, Int, Int]", MonoidalTests[Cokleisli[Option, Int, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Cokleisli[Option, Int, ?]", SerializableTests.serializable(Monoidal[Cokleisli[Option, Int, ?]]))
+  checkAll("Cokleisli[Option, Int, Int]", SemigroupalTests[Cokleisli[Option, Int, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Cokleisli[Option, Int, ?]", SerializableTests.serializable(Semigroupal[Cokleisli[Option, Int, ?]]))
 
   checkAll("Cokleisli[Option, Int, Int]", ApplicativeTests[Cokleisli[Option, Int, ?]].applicative[Int, Int, Int])
   checkAll("Applicative[Cokleisli[Option, Int, ?]", SerializableTests.serializable(Applicative[Cokleisli[Option, Int, ?]]))
diff --git a/tests/src/test/scala/cats/tests/ComposeTests.scala b/tests/src/test/scala/cats/tests/ComposeTests.scala
index 5aca2b5628c..132b5c82db3 100644
--- a/tests/src/test/scala/cats/tests/ComposeTests.scala
+++ b/tests/src/test/scala/cats/tests/ComposeTests.scala
@@ -2,7 +2,7 @@ package cats
 package tests
 
 import cats.data.{ NonEmptyList, NonEmptyVector, OneAnd }
-import cats.laws.discipline.{ ApplicativeTests, FoldableTests, MonoidalTests, SemigroupKTests, arbitrary, eq }, arbitrary._, eq._
+import cats.laws.discipline.{ ApplicativeTests, FoldableTests, SemigroupalTests, SemigroupKTests, arbitrary, eq }, arbitrary._, eq._
 import org.scalacheck.Arbitrary
 
 class ComposeTests extends CatsSuite {
@@ -16,7 +16,7 @@ class ComposeTests extends CatsSuite {
     // Applicative composition
 
     implicit val applicativeListVector: Applicative[Lambda[A => List[Vector[A]]]] = Applicative[List] compose Applicative[Vector]
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[Lambda[A => List[Vector[A]]]]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[Lambda[A => List[Vector[A]]]]
 
     checkAll("Applicative[Lambda[A => List[Vector[A]]]]", ApplicativeTests[Lambda[A => List[Vector[A]]]].applicative[Int, Int, Int])
   }
diff --git a/tests/src/test/scala/cats/tests/ConstTests.scala b/tests/src/test/scala/cats/tests/ConstTests.scala
index c52482425a4..59be7cf94d3 100644
--- a/tests/src/test/scala/cats/tests/ConstTests.scala
+++ b/tests/src/test/scala/cats/tests/ConstTests.scala
@@ -10,10 +10,10 @@ import cats.laws.discipline.arbitrary.{constArbitrary, oneAndArbitrary}
 
 class ConstTests extends CatsSuite {
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Const[String, ?]](Const.constTraverse)
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Const[String, ?]](Const.constTraverse)
 
-  checkAll("Const[String, Int]", MonoidalTests[Const[String, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Const[String, ?]]", SerializableTests.serializable(Monoidal[Const[String, ?]]))
+  checkAll("Const[String, Int]", SemigroupalTests[Const[String, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Const[String, ?]]", SerializableTests.serializable(Semigroupal[Const[String, ?]]))
 
   checkAll("Const[String, Int]", ApplicativeTests[Const[String, ?]].applicative[Int, Int, Int])
   checkAll("Applicative[Const[String, ?]]", SerializableTests.serializable(Applicative[Const[String, ?]]))
@@ -24,7 +24,7 @@ class ConstTests extends CatsSuite {
   // Get Apply[Const[C : Semigroup, ?]], not Applicative[Const[C : Monoid, ?]]
   {
     implicit def nonEmptyListSemigroup[A]: Semigroup[NonEmptyList[A]] = SemigroupK[NonEmptyList].algebra
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[Const[NonEmptyList[String], ?]](Const.constContravariant)
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[Const[NonEmptyList[String], ?]](Const.constContravariant)
     checkAll("Apply[Const[NonEmptyList[String], Int]]", ApplyTests[Const[NonEmptyList[String], ?]].apply[Int, Int, Int])
     checkAll("Apply[Const[NonEmptyList[String], ?]]", SerializableTests.serializable(Apply[Const[NonEmptyList[String], ?]]))
   }
diff --git a/tests/src/test/scala/cats/tests/EitherTests.scala b/tests/src/test/scala/cats/tests/EitherTests.scala
index 5c4a015968e..8dca767d8a1 100644
--- a/tests/src/test/scala/cats/tests/EitherTests.scala
+++ b/tests/src/test/scala/cats/tests/EitherTests.scala
@@ -1,16 +1,16 @@
 package cats
 package tests
 
-import cats.laws.discipline.{TraverseTests, MonadTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{TraverseTests, MonadTests, SerializableTests, SemigroupalTests}
 import cats.laws.discipline.eq._
 import algebra.laws.OrderLaws
 
 class EitherTests extends CatsSuite {
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Either[Int, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Either[Int, ?]]
 
-  checkAll("Either[Int, Int]", MonoidalTests[Either[Int, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Either[Int, ?]]", SerializableTests.serializable(Monoidal[Either[Int, ?]]))
+  checkAll("Either[Int, Int]", SemigroupalTests[Either[Int, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Either[Int, ?]]", SerializableTests.serializable(Semigroupal[Either[Int, ?]]))
 
   checkAll("Either[Int, Int]", MonadTests[Either[Int, ?]].monad[Int, Int, Int])
   checkAll("Monad[Either[Int, ?]]", SerializableTests.serializable(Monad[Either[Int, ?]]))
diff --git a/tests/src/test/scala/cats/tests/EvalTests.scala b/tests/src/test/scala/cats/tests/EvalTests.scala
index 7c40556908c..a02a5522802 100644
--- a/tests/src/test/scala/cats/tests/EvalTests.scala
+++ b/tests/src/test/scala/cats/tests/EvalTests.scala
@@ -3,7 +3,7 @@ package tests
 
 import scala.math.min
 import cats.laws.ComonadLaws
-import cats.laws.discipline.{MonoidalTests, BimonadTests, SerializableTests}
+import cats.laws.discipline.{SemigroupalTests, BimonadTests, SerializableTests}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
 import algebra.laws.{GroupLaws, OrderLaws}
@@ -92,7 +92,7 @@ class EvalTests extends CatsSuite {
   }
 
   {
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[Eval]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[Eval]
     checkAll("Eval[Int]", BimonadTests[Eval].bimonad[Int, Int, Int])
   }
   checkAll("Bimonad[Eval]", SerializableTests.serializable(Bimonad[Eval]))
diff --git a/tests/src/test/scala/cats/tests/FreeApplicativeTests.scala b/tests/src/test/scala/cats/tests/FreeApplicativeTests.scala
index aae6fdf5ca0..c594eb6b4f9 100644
--- a/tests/src/test/scala/cats/tests/FreeApplicativeTests.scala
+++ b/tests/src/test/scala/cats/tests/FreeApplicativeTests.scala
@@ -3,7 +3,7 @@ package tests
 
 import cats.arrow.NaturalTransformation
 import cats.free.FreeApplicative
-import cats.laws.discipline.{MonoidalTests, ApplicativeTests, SerializableTests}
+import cats.laws.discipline.{SemigroupalTests, ApplicativeTests, SerializableTests}
 import cats.laws.discipline.eq.tuple3Eq
 import cats.data.Const
 
@@ -24,7 +24,7 @@ class FreeApplicativeTests extends CatsSuite {
       }
     }
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[FreeApplicative[Option, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[FreeApplicative[Option, ?]]
 
   checkAll("FreeApplicative[Option, ?]", ApplicativeTests[FreeApplicative[Option, ?]].applicative[Int, Int, Int])
   checkAll("Monad[FreeApplicative[Option, ?]]", SerializableTests.serializable(Applicative[FreeApplicative[Option, ?]]))
diff --git a/tests/src/test/scala/cats/tests/FreeTests.scala b/tests/src/test/scala/cats/tests/FreeTests.scala
index 96d58866f00..66d73a926d0 100644
--- a/tests/src/test/scala/cats/tests/FreeTests.scala
+++ b/tests/src/test/scala/cats/tests/FreeTests.scala
@@ -3,7 +3,7 @@ package tests
 
 import cats.arrow.NaturalTransformation
 import cats.free.{Free, Trampoline}
-import cats.laws.discipline.{MonoidalTests, MonadTests, SerializableTests}
+import cats.laws.discipline.{SemigroupalTests, MonadTests, SerializableTests}
 import cats.laws.discipline.eq._
 import cats.laws.discipline.arbitrary.function0Arbitrary
 import org.scalacheck.{Arbitrary, Gen}
@@ -12,7 +12,7 @@ import Arbitrary.{arbitrary, arbFunction1}
 class FreeTests extends CatsSuite {
   import FreeTests._
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Free[Option, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Free[Option, ?]]
 
   checkAll("Free[Option, ?]", MonadTests[Free[Option, ?]].monad[Int, Int, Int])
   checkAll("Monad[Free[Option, ?]]", SerializableTests.serializable(Monad[Free[Option, ?]]))
diff --git a/tests/src/test/scala/cats/tests/FuncTests.scala b/tests/src/test/scala/cats/tests/FuncTests.scala
index 65651965828..ff9afdff452 100644
--- a/tests/src/test/scala/cats/tests/FuncTests.scala
+++ b/tests/src/test/scala/cats/tests/FuncTests.scala
@@ -14,10 +14,10 @@ class FuncTests extends CatsSuite {
   implicit def appFuncEq[F[_], A, B](implicit A: Arbitrary[A], FB: Eq[F[B]]): Eq[AppFunc[F, A, B]] =
     Eq.by[AppFunc[F, A, B], A => F[B]](_.run)
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Func[Option, Int, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Func[Option, Int, ?]]
 
-  checkAll("Func[Option, Int, Int]", MonoidalTests[Func[Option, Int, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Func[Option, Int, ?]]", SerializableTests.serializable(Monoidal[Func[Option, Int, ?]]))
+  checkAll("Func[Option, Int, Int]", SemigroupalTests[Func[Option, Int, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Func[Option, Int, ?]]", SerializableTests.serializable(Semigroupal[Func[Option, Int, ?]]))
 
   {
     implicit val funcApp = Func.funcApplicative[Option, Int]
@@ -39,7 +39,7 @@ class FuncTests extends CatsSuite {
 
   {
     implicit val appFuncApp = AppFunc.appFuncApplicative[Option, Int]
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[AppFunc[Option, Int, ?]]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[AppFunc[Option, Int, ?]]
     checkAll("AppFunc[Option, Int, Int]", ApplicativeTests[AppFunc[Option, Int, ?]].applicative[Int, Int, Int])
     checkAll("Applicative[AppFunc[Option, Int, ?]]", SerializableTests.serializable(Applicative[AppFunc[Option, Int, ?]]))
   }
diff --git a/tests/src/test/scala/cats/tests/FunctionTests.scala b/tests/src/test/scala/cats/tests/FunctionTests.scala
index 185dc8eaab0..2a137fd1729 100644
--- a/tests/src/test/scala/cats/tests/FunctionTests.scala
+++ b/tests/src/test/scala/cats/tests/FunctionTests.scala
@@ -12,15 +12,15 @@ import algebra.laws.GroupLaws
 
 class FunctionTests extends CatsSuite {
 
-  checkAll("Function0[Int]", MonoidalTests[Function0].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Function0]", SerializableTests.serializable(Monoidal[Function0]))
+  checkAll("Function0[Int]", SemigroupalTests[Function0].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Function0]", SerializableTests.serializable(Semigroupal[Function0]))
 
   checkAll("Function0[Int]", BimonadTests[Function0].bimonad[Int, Int, Int])
   checkAll("Bimonad[Function0]", SerializableTests.serializable(Bimonad[Function0]))
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Function1[Int, ?]]
-  checkAll("Function1[Int, Int]", MonoidalTests[Function1[Int, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Function1[Int, ?]]", SerializableTests.serializable(Monoidal[Function1[Int, ?]]))
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Function1[Int, ?]]
+  checkAll("Function1[Int, Int]", SemigroupalTests[Function1[Int, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Function1[Int, ?]]", SerializableTests.serializable(Semigroupal[Function1[Int, ?]]))
 
   checkAll("Function1[Int, Int]", MonadReaderTests[Int => ?, Int].monadReader[Int, Int, Int])
   checkAll("MonadReader[Int => ?, Int]", SerializableTests.serializable(MonadReader[Int => ?, Int]))
diff --git a/tests/src/test/scala/cats/tests/IdTests.scala b/tests/src/test/scala/cats/tests/IdTests.scala
index 572f6d66996..0874f3fee68 100644
--- a/tests/src/test/scala/cats/tests/IdTests.scala
+++ b/tests/src/test/scala/cats/tests/IdTests.scala
@@ -6,7 +6,7 @@ import cats.laws.discipline._
 import cats.laws.discipline.eq.tuple3Eq
 
 class IdTests extends CatsSuite {
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Id]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Id]
 
   checkAll("Id[Int]", BimonadTests[Id].bimonad[Int, Int, Int])
   checkAll("Bimonad[Id]", SerializableTests.serializable(Bimonad[Id]))
diff --git a/tests/src/test/scala/cats/tests/IorTests.scala b/tests/src/test/scala/cats/tests/IorTests.scala
index d1c12437b1b..b4667885a9e 100644
--- a/tests/src/test/scala/cats/tests/IorTests.scala
+++ b/tests/src/test/scala/cats/tests/IorTests.scala
@@ -2,7 +2,7 @@ package cats
 package tests
 
 import cats.data.{Xor, Ior}
-import cats.laws.discipline.{BifunctorTests, TraverseTests, MonadTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{BifunctorTests, TraverseTests, MonadTests, SerializableTests, SemigroupalTests}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
 import org.scalacheck.Arbitrary
@@ -10,10 +10,10 @@ import org.scalacheck.Arbitrary._
 
 class IorTests extends CatsSuite {
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Ior[String, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Ior[String, ?]]
 
-  checkAll("Ior[String, Int]", MonoidalTests[Ior[String, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[String Ior ?]]", SerializableTests.serializable(Monoidal[String Ior ?]))
+  checkAll("Ior[String, Int]", SemigroupalTests[Ior[String, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[String Ior ?]]", SerializableTests.serializable(Semigroupal[String Ior ?]))
 
   checkAll("Ior[String, Int]", MonadTests[String Ior ?].monad[Int, Int, Int])
   checkAll("Monad[String Ior ?]]", SerializableTests.serializable(Monad[String Ior ?]))
diff --git a/tests/src/test/scala/cats/tests/KleisliTests.scala b/tests/src/test/scala/cats/tests/KleisliTests.scala
index 9c5f6cea221..6d09efda7fb 100644
--- a/tests/src/test/scala/cats/tests/KleisliTests.scala
+++ b/tests/src/test/scala/cats/tests/KleisliTests.scala
@@ -15,10 +15,10 @@ class KleisliTests extends CatsSuite {
   implicit def kleisliEq[F[_], A, B](implicit A: Arbitrary[A], FB: Eq[F[B]]): Eq[Kleisli[F, A, B]] =
     Eq.by[Kleisli[F, A, B], A => F[B]](_.run)
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Kleisli[Option, Int, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Kleisli[Option, Int, ?]]
 
-  checkAll("Kleisli[Option, Int, Int]", MonoidalTests[Kleisli[Option, Int, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Kleisli[Option, Int, ?]]", SerializableTests.serializable(Monoidal[Kleisli[Option, Int, ?]]))
+  checkAll("Kleisli[Option, Int, Int]", SemigroupalTests[Kleisli[Option, Int, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Kleisli[Option, Int, ?]]", SerializableTests.serializable(Semigroupal[Kleisli[Option, Int, ?]]))
 
   {
     implicit val kleisliArrow = Kleisli.kleisliArrow[Option]
diff --git a/tests/src/test/scala/cats/tests/ListTests.scala b/tests/src/test/scala/cats/tests/ListTests.scala
index 9f780005428..899781c162b 100644
--- a/tests/src/test/scala/cats/tests/ListTests.scala
+++ b/tests/src/test/scala/cats/tests/ListTests.scala
@@ -2,14 +2,14 @@ package cats
 package tests
 
 import cats.data.NonEmptyList
-import cats.laws.discipline.{TraverseTests, CoflatMapTests, MonadCombineTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{TraverseTests, CoflatMapTests, MonadCombineTests, SerializableTests, SemigroupalTests}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
 
 class ListTests extends CatsSuite {
 
-  checkAll("List[Int]", MonoidalTests[List].monoidal[Int, Int, Int])
-  checkAll("Monoidal[List]", SerializableTests.serializable(Monoidal[List]))
+  checkAll("List[Int]", SemigroupalTests[List].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[List]", SerializableTests.serializable(Semigroupal[List]))
 
   checkAll("List[Int]", CoflatMapTests[List].coflatMap[Int, Int, Int])
   checkAll("CoflatMap[List]", SerializableTests.serializable(CoflatMap[List]))
diff --git a/tests/src/test/scala/cats/tests/MapTests.scala b/tests/src/test/scala/cats/tests/MapTests.scala
index ba6d77240bb..0b1a026f8ed 100644
--- a/tests/src/test/scala/cats/tests/MapTests.scala
+++ b/tests/src/test/scala/cats/tests/MapTests.scala
@@ -1,14 +1,14 @@
 package cats
 package tests
 
-import cats.laws.discipline.{TraverseTests, FlatMapTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{TraverseTests, FlatMapTests, SerializableTests, SemigroupalTests}
 import cats.laws.discipline.eq._
 
 class MapTests extends CatsSuite {
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Map[Int, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Map[Int, ?]]
 
-  checkAll("Map[Int, Int]", MonoidalTests[Map[Int, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Map[Int, ?]]", SerializableTests.serializable(Monoidal[Map[Int, ?]]))
+  checkAll("Map[Int, Int]", SemigroupalTests[Map[Int, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Map[Int, ?]]", SerializableTests.serializable(Semigroupal[Map[Int, ?]]))
 
   checkAll("Map[Int, Int]", FlatMapTests[Map[Int, ?]].flatMap[Int, Int, Int])
   checkAll("FlatMap[Map[Int, ?]]", SerializableTests.serializable(FlatMap[Map[Int, ?]]))
diff --git a/tests/src/test/scala/cats/tests/OneAndTests.scala b/tests/src/test/scala/cats/tests/OneAndTests.scala
index 0b6f5c3ea0d..f14c1bd2610 100644
--- a/tests/src/test/scala/cats/tests/OneAndTests.scala
+++ b/tests/src/test/scala/cats/tests/OneAndTests.scala
@@ -4,7 +4,7 @@ package tests
 import algebra.laws.{GroupLaws, OrderLaws}
 
 import cats.data.{NonEmptyList, OneAnd}
-import cats.laws.discipline.{ComonadTests, FunctorTests, SemigroupKTests, FoldableTests, MonadTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{ComonadTests, FunctorTests, SemigroupKTests, FoldableTests, MonadTests, SerializableTests, SemigroupalTests}
 import cats.laws.discipline.arbitrary.{evalArbitrary, oneAndArbitrary}
 import cats.laws.discipline.eq._
 
@@ -13,13 +13,13 @@ import scala.util.Random
 class OneAndTests extends CatsSuite {
   checkAll("OneAnd[List, Int]", OrderLaws[OneAnd[List, Int]].eqv)
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[OneAnd[ListWrapper, ?]](OneAnd.oneAndFunctor(ListWrapper.functor))
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[OneAnd[ListWrapper, ?]](OneAnd.oneAndFunctor(ListWrapper.functor))
 
   // Test instances that have more general constraints
   {
     implicit val monadCombine = ListWrapper.monadCombine
-    checkAll("OneAnd[ListWrapper, Int]", MonoidalTests[OneAnd[ListWrapper, ?]].monoidal[Int, Int, Int])
-    checkAll("Monoidal[OneAnd[ListWrapper, A]]", SerializableTests.serializable(Monoidal[OneAnd[ListWrapper, ?]]))
+    checkAll("OneAnd[ListWrapper, Int]", SemigroupalTests[OneAnd[ListWrapper, ?]].semigroupal[Int, Int, Int])
+    checkAll("Semigroupal[OneAnd[ListWrapper, A]]", SerializableTests.serializable(Semigroupal[OneAnd[ListWrapper, ?]]))
   }
 
   {
@@ -49,7 +49,7 @@ class OneAndTests extends CatsSuite {
     implicitly[Comonad[NonEmptyList]]
   }
 
-  implicit val iso2 = MonoidalTests.Isomorphisms.invariant[OneAnd[List, ?]]
+  implicit val iso2 = SemigroupalTests.Isomorphisms.invariant[OneAnd[List, ?]]
 
   checkAll("NonEmptyList[Int]", MonadTests[NonEmptyList].monad[Int, Int, Int])
   checkAll("Monad[NonEmptyList[A]]", SerializableTests.serializable(Monad[NonEmptyList]))
diff --git a/tests/src/test/scala/cats/tests/OptionTTests.scala b/tests/src/test/scala/cats/tests/OptionTTests.scala
index 12f50c1ec0c..c82c284741b 100644
--- a/tests/src/test/scala/cats/tests/OptionTTests.scala
+++ b/tests/src/test/scala/cats/tests/OptionTTests.scala
@@ -1,8 +1,8 @@
 package cats.tests
 
-import cats.{Applicative, Id, Monad, Monoidal, Show}
+import cats.{Applicative, Id, Monad, Semigroupal, Show}
 import cats.data.{OptionT, Validated, Xor}
-import cats.laws.discipline.{ApplicativeTests, FunctorTests, MonadCombineTests, SerializableTests, MonoidalTests, MonadTests}
+import cats.laws.discipline.{ApplicativeTests, FunctorTests, MonadCombineTests, SerializableTests, SemigroupalTests, MonadTests}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
 import org.scalacheck.{Arbitrary, Gen}
@@ -118,10 +118,10 @@ class OptionTTests extends CatsSuite {
     }
   }
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[OptionT[List, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[OptionT[List, ?]]
 
-  checkAll("OptionT[List, Int]", MonoidalTests[OptionT[List, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[OptionT[List, ?]]", SerializableTests.serializable(Monoidal[OptionT[List, ?]]))
+  checkAll("OptionT[List, Int]", SemigroupalTests[OptionT[List, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[OptionT[List, ?]]", SerializableTests.serializable(Semigroupal[OptionT[List, ?]]))
 
   test("liftF") {
     forAll { (xs: List[Int]) =>
diff --git a/tests/src/test/scala/cats/tests/OptionTests.scala b/tests/src/test/scala/cats/tests/OptionTests.scala
index 77864b6d251..0f4945ba8c1 100644
--- a/tests/src/test/scala/cats/tests/OptionTests.scala
+++ b/tests/src/test/scala/cats/tests/OptionTests.scala
@@ -2,12 +2,12 @@ package cats
 package tests
 
 import cats.laws.{ApplicativeLaws, CoflatMapLaws, FlatMapLaws, MonadLaws}
-import cats.laws.discipline.{TraverseTests, CoflatMapTests, MonadCombineTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{TraverseTests, CoflatMapTests, MonadCombineTests, SerializableTests, SemigroupalTests}
 import cats.laws.discipline.eq._
 
 class OptionTests extends CatsSuite {
-  checkAll("Option[Int]", MonoidalTests[Option].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Option]", SerializableTests.serializable(Monoidal[Option]))
+  checkAll("Option[Int]", SemigroupalTests[Option].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Option]", SerializableTests.serializable(Semigroupal[Option]))
 
   checkAll("Option[Int]", CoflatMapTests[Option].coflatMap[Int, Int, Int])
   checkAll("CoflatMap[Option]", SerializableTests.serializable(CoflatMap[Option]))
diff --git a/tests/src/test/scala/cats/tests/ProdTests.scala b/tests/src/test/scala/cats/tests/ProdTests.scala
index 28d40f8a622..2abb98c7644 100644
--- a/tests/src/test/scala/cats/tests/ProdTests.scala
+++ b/tests/src/test/scala/cats/tests/ProdTests.scala
@@ -8,9 +8,9 @@ import cats.laws.discipline.eq._
 import org.scalacheck.Arbitrary
 
 class ProdTests extends CatsSuite {
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Prod[Option, List, ?]]
-  checkAll("Prod[Option, List, Int]", MonoidalTests[Lambda[X => Prod[Option, List, X]]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Prod[Option, List, Int]]", SerializableTests.serializable(Monoidal[Lambda[X => Prod[Option, List, X]]]))
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Prod[Option, List, ?]]
+  checkAll("Prod[Option, List, Int]", SemigroupalTests[Lambda[X => Prod[Option, List, X]]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Prod[Option, List, Int]]", SerializableTests.serializable(Semigroupal[Lambda[X => Prod[Option, List, X]]]))
 
   checkAll("Prod[Option, List, Int]", AlternativeTests[Lambda[X => Prod[Option, List, X]]].alternative[Int, Int, Int])
   checkAll("Alternative[Prod[Option, List, Int]]", SerializableTests.serializable(Alternative[Lambda[X => Prod[Option, List, X]]]))
diff --git a/tests/src/test/scala/cats/tests/StateTTests.scala b/tests/src/test/scala/cats/tests/StateTTests.scala
index 257f89e46e4..1798851cb2e 100644
--- a/tests/src/test/scala/cats/tests/StateTTests.scala
+++ b/tests/src/test/scala/cats/tests/StateTTests.scala
@@ -1,7 +1,7 @@
 package cats
 package tests
 
-import cats.laws.discipline.{MonoidalTests, MonadStateTests, MonoidKTests, SerializableTests}
+import cats.laws.discipline.{SemigroupalTests, MonadStateTests, MonoidKTests, SerializableTests}
 import cats.state.{State, StateT}
 import cats.laws.discipline.eq._
 import cats.laws.discipline.arbitrary._
@@ -28,7 +28,7 @@ class StateTTests extends CatsSuite {
     }
   }
 
-  test("Monoidal syntax is usable on State") {
+  test("Semigroupal syntax is usable on State") {
     val x = add1 *> add1
     x.runS(0).value should === (2)
   }
@@ -95,13 +95,13 @@ class StateTTests extends CatsSuite {
   }
 
   {
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[StateT[Option, Int, ?]]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[StateT[Option, Int, ?]]
     checkAll("StateT[Option, Int, Int]", MonadStateTests[StateT[Option, Int, ?], Int].monadState[Int, Int, Int])
     checkAll("MonadState[StateT[Option, ?, ?], Int]", SerializableTests.serializable(MonadState[StateT[Option, Int, ?], Int]))
   }
 
   {
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[State[Long, ?]]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[State[Long, ?]]
     checkAll("State[Long, ?]", MonadStateTests[State[Long, ?], Long].monadState[Int, Int, Int])
     checkAll("MonadState[State[Long, ?], Long]", SerializableTests.serializable(MonadState[State[Long, ?], Long]))
   }
diff --git a/tests/src/test/scala/cats/tests/StreamTests.scala b/tests/src/test/scala/cats/tests/StreamTests.scala
index 828fed2ddbb..57a3e652c5d 100644
--- a/tests/src/test/scala/cats/tests/StreamTests.scala
+++ b/tests/src/test/scala/cats/tests/StreamTests.scala
@@ -1,12 +1,12 @@
 package cats
 package tests
 
-import cats.laws.discipline.{CoflatMapTests, MonadCombineTests, SerializableTests, TraverseTests, MonoidalTests}
+import cats.laws.discipline.{CoflatMapTests, MonadCombineTests, SerializableTests, TraverseTests, SemigroupalTests}
 import cats.laws.discipline.eq.tuple3Eq
 
 class StreamTests extends CatsSuite {
-  checkAll("Stream[Int]", MonoidalTests[Stream].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Stream]", SerializableTests.serializable(Monoidal[Stream]))
+  checkAll("Stream[Int]", SemigroupalTests[Stream].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Stream]", SerializableTests.serializable(Semigroupal[Stream]))
 
   checkAll("Stream[Int]", CoflatMapTests[Stream].coflatMap[Int, Int, Int])
   checkAll("CoflatMap[Stream]", SerializableTests.serializable(CoflatMap[Stream]))
diff --git a/tests/src/test/scala/cats/tests/StreamingTTests.scala b/tests/src/test/scala/cats/tests/StreamingTTests.scala
index 753bdae8289..3fc1a42c2c4 100644
--- a/tests/src/test/scala/cats/tests/StreamingTTests.scala
+++ b/tests/src/test/scala/cats/tests/StreamingTTests.scala
@@ -4,14 +4,14 @@ package tests
 import algebra.laws.OrderLaws
 
 import cats.data.{Streaming, StreamingT}
-import cats.laws.discipline.{MonoidalTests, CoflatMapTests, MonadCombineTests, SerializableTests}
+import cats.laws.discipline.{SemigroupalTests, CoflatMapTests, MonadCombineTests, SerializableTests}
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
 
 class StreamingTTests extends CatsSuite {
 
   {
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[StreamingT[Eval, ?]]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[StreamingT[Eval, ?]]
     checkAll("StreamingT[Eval, ?]", MonadCombineTests[StreamingT[Eval, ?]].monadCombine[Int, Int, Int])
     checkAll("StreamingT[Eval, ?]", CoflatMapTests[StreamingT[Eval, ?]].coflatMap[Int, Int, Int])
     checkAll("StreamingT[Eval, Int]", OrderLaws[StreamingT[Eval, Int]].order)
@@ -19,7 +19,7 @@ class StreamingTTests extends CatsSuite {
   }
 
   {
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[StreamingT[Option, ?]]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[StreamingT[Option, ?]]
     checkAll("StreamingT[Option, ?]", MonadCombineTests[StreamingT[Option, ?]].monadCombine[Int, Int, Int])
     checkAll("StreamingT[Option, ?]", CoflatMapTests[StreamingT[Option, ?]].coflatMap[Int, Int, Int])
     checkAll("StreamingT[Option, Int]", OrderLaws[StreamingT[Option, Int]].order)
@@ -27,7 +27,7 @@ class StreamingTTests extends CatsSuite {
   }
 
   {
-    implicit val iso = MonoidalTests.Isomorphisms.invariant[StreamingT[List, ?]]
+    implicit val iso = SemigroupalTests.Isomorphisms.invariant[StreamingT[List, ?]]
     checkAll("StreamingT[List, ?]", MonadCombineTests[StreamingT[List, ?]].monadCombine[Int, Int, Int])
     checkAll("StreamingT[List, ?]", CoflatMapTests[StreamingT[List, ?]].coflatMap[Int, Int, Int])
     checkAll("StreamingT[List, Int]", OrderLaws[StreamingT[List, Int]].order)
diff --git a/tests/src/test/scala/cats/tests/StreamingTests.scala b/tests/src/test/scala/cats/tests/StreamingTests.scala
index 3097c5ce355..d93edf234e9 100644
--- a/tests/src/test/scala/cats/tests/StreamingTests.scala
+++ b/tests/src/test/scala/cats/tests/StreamingTests.scala
@@ -6,12 +6,12 @@ import algebra.laws.OrderLaws
 import cats.data.Streaming
 import cats.laws.discipline.arbitrary._
 import cats.laws.discipline.eq._
-import cats.laws.discipline.{TraverseTests, CoflatMapTests, MonadCombineTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{TraverseTests, CoflatMapTests, MonadCombineTests, SerializableTests, SemigroupalTests}
 import org.scalacheck.{Arbitrary, Gen}
 
 class StreamingTests extends CatsSuite {
-  checkAll("Streaming[Int]", MonoidalTests[Streaming].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Streaming]", SerializableTests.serializable(Monoidal[Streaming]))
+  checkAll("Streaming[Int]", SemigroupalTests[Streaming].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Streaming]", SerializableTests.serializable(Semigroupal[Streaming]))
 
   checkAll("Streaming[Int]", CoflatMapTests[Streaming].coflatMap[Int, Int, Int])
   checkAll("CoflatMap[Streaming]", SerializableTests.serializable(CoflatMap[Streaming]))
diff --git a/tests/src/test/scala/cats/tests/ValidatedTests.scala b/tests/src/test/scala/cats/tests/ValidatedTests.scala
index ea575a8d860..3e42ca6e087 100644
--- a/tests/src/test/scala/cats/tests/ValidatedTests.scala
+++ b/tests/src/test/scala/cats/tests/ValidatedTests.scala
@@ -3,7 +3,7 @@ package tests
 
 import cats.data.{NonEmptyList, Validated, ValidatedNel, Xor}
 import cats.data.Validated.{Valid, Invalid}
-import cats.laws.discipline.{BifunctorTests, TraverseTests, ApplicativeTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{BifunctorTests, TraverseTests, ApplicativeTests, SerializableTests, SemigroupalTests}
 import org.scalacheck.{Gen, Arbitrary}
 import org.scalacheck.Arbitrary._
 import cats.laws.discipline.arbitrary._
@@ -13,9 +13,9 @@ import algebra.laws.{OrderLaws, GroupLaws}
 import scala.util.Try
 
 class ValidatedTests extends CatsSuite {
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Validated[String, ?]]
-  checkAll("Validated[String, Int]", MonoidalTests[Validated[String,?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Validated[String,?]]", SerializableTests.serializable(Monoidal[Validated[String,?]]))
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Validated[String, ?]]
+  checkAll("Validated[String, Int]", SemigroupalTests[Validated[String,?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Validated[String,?]]", SerializableTests.serializable(Semigroupal[Validated[String,?]]))
 
   checkAll("Validated[String, Int]", ApplicativeTests[Validated[String,?]].applicative[Int, Int, Int])
   checkAll("Validated[?, ?]", BifunctorTests[Validated].bifunctor[Int, Int, Int, Int, Int, Int])
diff --git a/tests/src/test/scala/cats/tests/VectorTests.scala b/tests/src/test/scala/cats/tests/VectorTests.scala
index 423e101a155..05acd0a87a4 100644
--- a/tests/src/test/scala/cats/tests/VectorTests.scala
+++ b/tests/src/test/scala/cats/tests/VectorTests.scala
@@ -1,12 +1,12 @@
 package cats
 package tests
 
-import cats.laws.discipline.{MonadCombineTests, SerializableTests, TraverseTests, MonoidalTests}
+import cats.laws.discipline.{MonadCombineTests, SerializableTests, TraverseTests, SemigroupalTests}
 import cats.laws.discipline.eq.tuple3Eq
 
 class VectorTests extends CatsSuite {
-  checkAll("Vector[Int]", MonoidalTests[Vector].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Vector]", SerializableTests.serializable(Monoidal[Vector]))
+  checkAll("Vector[Int]", SemigroupalTests[Vector].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Vector]", SerializableTests.serializable(Semigroupal[Vector]))
 
   checkAll("Vector[Int]", MonadCombineTests[Vector].monadCombine[Int, Int, Int])
   checkAll("MonadCombine[Vector]", SerializableTests.serializable(MonadCombine[Vector]))
diff --git a/tests/src/test/scala/cats/tests/WordCountTest.scala b/tests/src/test/scala/cats/tests/WordCountTest.scala
index 8324349a380..15b2b19b249 100644
--- a/tests/src/test/scala/cats/tests/WordCountTest.scala
+++ b/tests/src/test/scala/cats/tests/WordCountTest.scala
@@ -12,7 +12,7 @@ class WordCountTest extends CatsSuite {
   test("wordcount") {
     import cats.state.State.{ get, set }
     val text = "Faith, I must leave thee, love, and shortly too.\nMy operant powers their functions leave to do.\n".toList
-    // A type alias to treat Int as monoidal applicative
+    // A type alias to treat Int as semigroupal applicative
     type Count[A] = Const[Int, A]
     // Tye type parameter to Count is ceremonial, so hardcode it to Unit
     def liftInt(i: Int): Count[Unit] = Const(i)
diff --git a/tests/src/test/scala/cats/tests/WriterTTests.scala b/tests/src/test/scala/cats/tests/WriterTTests.scala
index d214a08899d..ec1c5f05bf2 100644
--- a/tests/src/test/scala/cats/tests/WriterTTests.scala
+++ b/tests/src/test/scala/cats/tests/WriterTTests.scala
@@ -80,7 +80,7 @@ class WriterTTests extends CatsSuite {
     checkAll("Bifunctor[WriterT[ListWrapper, ?, ?]]", SerializableTests.serializable(Bifunctor[WriterT[ListWrapper, ?, ?]]))
   }
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[WriterT[ListWrapper, ListWrapper[Int], ?]](WriterT.writerTFunctor(ListWrapper.functor))
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[WriterT[ListWrapper, ListWrapper[Int], ?]](WriterT.writerTFunctor(ListWrapper.functor))
 
   // We have varying instances available depending on `F` and `L`.
   // We also battle some inference issues with `Id`.
diff --git a/tests/src/test/scala/cats/tests/XorTTests.scala b/tests/src/test/scala/cats/tests/XorTTests.scala
index c24f1561ff0..b5548a15b99 100644
--- a/tests/src/test/scala/cats/tests/XorTTests.scala
+++ b/tests/src/test/scala/cats/tests/XorTTests.scala
@@ -11,7 +11,7 @@ import algebra.laws.OrderLaws
 class XorTTests extends CatsSuite {
   implicit val eq0 = XorT.xorTEq[List, String, String Xor Int]
   implicit val eq1 = XorT.xorTEq[XorT[List, String, ?], String, Int](eq0)
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[XorT[List, String, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[XorT[List, String, ?]]
   checkAll("XorT[List, String, Int]", MonadErrorTests[XorT[List, String, ?], String].monadError[Int, Int, Int])
   checkAll("MonadError[XorT[List, ?, ?]]", SerializableTests.serializable(MonadError[XorT[List, String, ?], String]))
   checkAll("XorT[List, String, Int]", MonoidKTests[XorT[List, String, ?]].monoidK[Int])
diff --git a/tests/src/test/scala/cats/tests/XorTests.scala b/tests/src/test/scala/cats/tests/XorTests.scala
index e8aec6c09b6..853fb20b4bc 100644
--- a/tests/src/test/scala/cats/tests/XorTests.scala
+++ b/tests/src/test/scala/cats/tests/XorTests.scala
@@ -4,7 +4,7 @@ package tests
 import cats.data.{NonEmptyList, Xor, XorT}
 import cats.data.Xor._
 import cats.laws.discipline.arbitrary._
-import cats.laws.discipline.{BifunctorTests, TraverseTests, MonadErrorTests, SerializableTests, MonoidalTests}
+import cats.laws.discipline.{BifunctorTests, TraverseTests, MonadErrorTests, SerializableTests, SemigroupalTests}
 import cats.laws.discipline.eq.tuple3Eq
 import algebra.laws.{GroupLaws, OrderLaws}
 import org.scalacheck.{Arbitrary, Gen}
@@ -15,10 +15,10 @@ import scala.util.Try
 class XorTests extends CatsSuite {
   checkAll("Xor[String, Int]", GroupLaws[Xor[String, Int]].monoid)
 
-  implicit val iso = MonoidalTests.Isomorphisms.invariant[Xor[String, ?]]
+  implicit val iso = SemigroupalTests.Isomorphisms.invariant[Xor[String, ?]]
 
-  checkAll("Xor[String, Int]", MonoidalTests[Xor[String, ?]].monoidal[Int, Int, Int])
-  checkAll("Monoidal[Xor, ?]", SerializableTests.serializable(Monoidal[Xor[String, ?]]))
+  checkAll("Xor[String, Int]", SemigroupalTests[Xor[String, ?]].semigroupal[Int, Int, Int])
+  checkAll("Semigroupal[Xor, ?]", SerializableTests.serializable(Semigroupal[Xor[String, ?]]))
 
   checkAll("Xor[String, NonEmptyList[Int]]", GroupLaws[Xor[String, NonEmptyList[Int]]].semigroup)