diff --git a/README.md b/README.md
index bc99392..bf1d6a9 100644
--- a/README.md
+++ b/README.md
@@ -114,8 +114,11 @@ values, that can be much faster.
 
 ## Changelog
 
-version 0.3.0.0 (2024-11-06):
+version 0.3.1.0 (2024-11-06):
 * More types of products
+* Stuff to do permutations (not only monotone ones)
+* New LStar variant, which can learn equivariant (wrt permutations) languages
+  with fewer queries. But it is slower.
 
 version 0.2.3.0 (2024-11-05):
 * Updates the testing and benchmarking framework.
diff --git a/app/ExampleAutomata.hs b/app/ExampleAutomata.hs
index 8502f44..526731e 100644
--- a/app/ExampleAutomata.hs
+++ b/app/ExampleAutomata.hs
@@ -1,6 +1,7 @@
 {-# language DeriveGeneric #-}
 {-# language DerivingVia #-}
 {-# language FlexibleContexts #-}
+{-# language ImportQualifiedPost #-}
 {-# language RecordWildCards #-}
 {-# language UndecidableInstances #-}
 {-# OPTIONS_GHC -Wno-missing-signatures #-}
@@ -10,14 +11,16 @@ module ExampleAutomata
   , module Automata
   ) where
 
-import Nominal hiding (product)
 import Automata
+import EquivariantMap qualified as Map
+import EquivariantSet qualified as Set
 import IO
+import Nominal hiding (product)
 import OrbitList
-import qualified EquivariantMap as Map
-import qualified EquivariantSet as Set
+import Permutable
 
 import Data.Foldable (fold)
+import Data.Map.Strict qualified as Data.Map
 import GHC.Generics (Generic)
 import Prelude as P hiding (map, product, words, filter, foldr)
 
@@ -69,6 +72,11 @@ data FifoA = Put Atom | Get Atom
   deriving (Eq, Ord, Show, Generic)
   deriving Nominal via Generically FifoA
 
+-- TODO: find a generic way to derive this.
+instance Permutable FifoA where
+  act (Permuted (Perm m) (Put p)) = Put $ Data.Map.findWithDefault p p m
+  act (Permuted (Perm m) (Get p)) = Get $ Data.Map.findWithDefault p p m
+
 instance ToStr FifoA where
   toStr (Put a) = "Put " ++ toStr a
   toStr (Get a) = "Get " ++ toStr a
diff --git a/app/LStarPerm.hs b/app/LStarPerm.hs
new file mode 100644
index 0000000..6b6b29e
--- /dev/null
+++ b/app/LStarPerm.hs
@@ -0,0 +1,227 @@
+{-# LANGUAGE DerivingVia #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE GeneralizedNewtypeDeriving #-}
+{-# LANGUAGE PartialTypeSignatures #-}
+{-# LANGUAGE RecordWildCards #-}
+{-# LANGUAGE StandaloneDeriving #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# OPTIONS_GHC -Wno-partial-type-signatures #-}
+
+import Automata (Word)
+import ExampleAutomata
+import IO
+import Quotient
+import OrbitList
+import EquivariantMap (EquivariantMap(..), (!))
+import qualified EquivariantMap as Map
+import qualified EquivariantSet as Set
+import Nominal (Nominal, Orbit, Trivially(..))
+import Permutable
+
+import Data.List (tails)
+import Data.Maybe (catMaybes)
+import Control.Monad.State
+import System.IO (hFlush, stdout)
+import Prelude hiding (filter, null, elem, lookup, product, Word, map, take, init)
+
+newtype PermEquivariantMap k v = PEqMap { unPEqMap :: EquivariantMap k v }
+  deriving Nominal via Trivially (EquivariantMap k v)
+
+-- Defined by the join-semilattice structure of EquivariantMap, left biased.
+deriving instance Ord (Orbit k) => Monoid (PermEquivariantMap k v)
+deriving instance Ord (Orbit k) => Semigroup (PermEquivariantMap k v)
+
+lookupP :: (Permutable k, Nominal k, Nominal v, _) => k -> PermEquivariantMap k v -> Maybe v
+lookupP x (PEqMap m) = case catMaybes [Map.lookup (act px) m | px <- allPermuted x] of
+  [] -> Nothing
+  (v:_) -> Just v -- take first hit, maybe this is wrong? I guess for v ~ Bool it's fine?
+
+insertP :: (Nominal k, Nominal v, _) => k -> v -> PermEquivariantMap k v -> PermEquivariantMap k v
+insertP k v = PEqMap . Map.insert k v . unPEqMap
+
+(!~) :: (Permutable k, Nominal k, Nominal v, _) => PermEquivariantMap k v -> k -> v
+(!~) m k = case lookupP k m of
+  Just v -> v
+  Nothing -> error "Key not found (in PermEquivariantMap)"
+
+-- We use Lists, as they provide a bit more laziness
+type Rows a    = OrbitList (Word a)
+type Columns a = OrbitList (Word a)
+type Table a   = PermEquivariantMap (Word a) Bool
+
+
+-- Utility functions
+exists f = not . null . filter f
+forAll f = null . filter (not . f)
+ext p a = p <> [a]
+
+equalRows :: _ => Word a -> Word a -> Columns a -> Table a -> Bool
+equalRows t0 s0 suffs table =
+  forAll (\((t, s), e) -> lookupP (s ++ e) table == lookupP (t ++ e) table) $ product (singleOrbit (t0, s0)) suffs
+
+closed :: _ => Word a -> Rows a -> Columns a -> Table a -> Bool
+closed t prefs suffs table =
+  exists (\(t, s) -> equalRows t s suffs table) (leftProduct (singleOrbit t) prefs)
+
+nonClosedness :: _ => Rows a -> Rows a -> Columns a -> Table a -> Rows a
+nonClosedness prefs prefsExt suffs table =
+  filter (\t -> not $ closed t prefs suffs table) prefsExt
+
+inconsistencies :: _ => Rows a -> Columns a -> Table a -> OrbitList a -> OrbitList ((Word a, Word a), (a, Word a))
+inconsistencies prefs suffs table alph =
+  filter (\((s, t), (a, e)) -> lookupP (s ++ (a:e)) table /= lookupP (t ++ (a:e)) table) candidatesExt
+  where
+    candidates = filter (\(s, t) -> s < t && equalRows s t suffs table) (product prefs prefs)
+    candidatesExt = product candidates (product alph suffs)
+
+
+-- Main state of the L* algorithm
+-- invariants: * prefs and prefsExt disjoint, without dups
+--             * prefsExt ordered
+--             * prefs and (prefs `union` prefsExt) prefix-closed
+--             * table defined on (prefs `union` prefsExt) * suffs
+data Observations a = Observations
+  { alph     :: OrbitList a
+  , prefs    :: Rows a
+  , prefsExt :: Rows a
+  , suffs    :: Columns a
+  , table    :: Table a
+  }
+
+-- input alphabet, inner monad, return value
+type LStar i m a = StateT (Observations i) m a
+
+-- First lookup, then membership query, also update the table
+ask mq (p, s) = do
+  Observations{..} <- get
+  let w = p ++ s
+  case lookupP w table of
+    Just b -> return (w, b)
+    Nothing -> do
+      b <- lift (mq w)
+      modify $ \o -> o { table = insertP w b table }
+      return (w, b)
+
+-- precondition: newPrefs is subset of prefExts
+addRows :: _ => Rows a -> (Word a -> m Bool) -> LStar a m ()
+addRows newPrefs mq = do
+  Observations{..} <- get
+  let newPrefsExt = productWith ext newPrefs alph
+      rect = product newPrefsExt suffs
+  _ <- mapM (ask mq) (OrbitList.toList rect)
+  modify $ \o -> o { prefs = prefs <> newPrefs
+                   , prefsExt = (prefsExt `minus` newPrefs) `union` newPrefsExt
+                   }
+  return ()
+
+-- precondition: newSuffs disjoint from suffs
+addCols :: _ => Columns a -> (Word a -> m Bool) -> LStar a m ()
+addCols newSuffs mq = do
+  Observations{..} <- get
+  let rect = product (prefs `union` prefsExt) newSuffs
+  _ <- mapM (ask mq) (OrbitList.toList rect)
+  modify $ \o -> o { suffs = suffs <> newSuffs }
+  return ()
+
+fillTable :: _ => (Word a -> m Bool) -> LStar a m ()
+fillTable mq = do
+  Observations{..} <- get
+  let rect = product (prefs `union` prefsExt) suffs
+  _ <- mapM (ask mq) (OrbitList.toList rect)
+  return ()
+
+-- This could be cleaned up
+learn :: _ => (Word a -> m Bool) -> (Automaton _ a -> m (Maybe (Word a))) -> LStar a m (Automaton _ a)
+learn mq eq = do
+  Observations{..} <- get
+  let ncl = nonClosedness prefs prefsExt suffs table
+      inc = inconsistencies prefs suffs table alph
+  case null ncl of
+    False -> do
+      -- If not closed, then add 1 orbit of rows. Then start from top
+      addRows (take 1 ncl) mq
+      learn mq eq
+    True -> do
+      -- Closed! Now we check consistency
+      case null inc of
+        False -> do
+          -- If not consistent, then add 1 orbit of columns. Then start from top
+          addCols (take 1 (map (uncurry (:) . snd) inc)) mq
+          learn mq eq
+        True -> do
+          -- Also consistent! Let's build a minimal automaton!
+          let (f, st, _) = quotientf 0 (\s t -> s == t || equalRows s t suffs table) prefs
+              trans = Map.fromList . toList . map (\(s, t) -> (s, f ! t)) . filter (\(s, t) -> equalRows s t suffs table) $ product prefsExt prefs
+              trans2 pa = if pa `elem` prefsExt then trans ! pa else f ! pa
+              hypothesis = Automaton
+                { states = map fst st
+                , initialState = f ! []
+                , acceptance = Map.fromList . toList . map (\p -> (f ! p, table !~ p)) $ prefs
+                , transition = Map.fromList . toList . map (\(p, a) -> ((f ! p, a), trans2 (ext p a))) $ product prefs alph
+                }
+              askCe = do
+                ce <- lift (eq hypothesis)
+                case ce of
+                  Nothing -> return hypothesis
+                  Just w -> do
+                    let b1 = accepts hypothesis w
+                    (_, b2) <- ask mq (w, [])
+                    -- Ignore false counterexamples
+                    case b1 == b2 of
+                      True -> askCe
+                      False -> do
+                        -- Add all suffixes of a counterexample
+                        let allSuffs = Set.fromList $ tails w
+                            newSuffs = allSuffs `Set.difference` Set.fromOrbitList suffs
+                        addCols (Set.toOrbitList newSuffs) mq
+                        learn mq eq
+          askCe
+
+
+-- Here is the teacher: just pose the queries in the terminal
+askMember :: _ => Word a -> IO Bool
+askMember w = do
+  putStrLn (toStr (MQ w))
+  hFlush stdout
+  a <- getLine
+  case a of
+    "Y" -> return True
+    "N" -> return False
+    _   -> askMember w
+
+askEquiv :: _ => Automaton q a -> IO (Maybe (Word a))
+askEquiv aut = do
+  putStr "EQ \""
+  putStr (toStr aut)
+  putStrLn "\""
+  hFlush stdout
+  a <- getLine
+  case a of
+    "Y"       -> return Nothing
+    'N':' ':w -> return $ Just (fst $ fromStr w)
+    _         -> askEquiv aut
+
+init alph = Observations
+  { alph = alph
+  , prefs = singleOrbit []
+  , prefsExt = productWith ext (singleOrbit []) alph
+  , suffs = singleOrbit[]
+  , table = mempty
+  }
+
+main :: IO ()
+main = do
+  putStrLn "ALPHABET"
+  hFlush stdout
+  alph <- getLine
+  case alph of
+    "ATOMS" -> do
+      aut <- evalStateT (fillTable askMember >> learn askMember askEquiv) (init rationals)
+      return ()
+    "FIFO" -> do
+      let alph = map Put rationals `union` map Get rationals
+      aut <- evalStateT (fillTable askMember >> learn askMember askEquiv) (init alph)
+      return ()
+    al -> do
+      putStr "Unknown alphabet "
+      putStrLn al
diff --git a/ons-hs.cabal b/ons-hs.cabal
index 4a70b22..fa73907 100644
--- a/ons-hs.cabal
+++ b/ons-hs.cabal
@@ -29,6 +29,7 @@ library
     Nominal.Class,
     Nominal.Products,
     OrbitList,
+    Permutable,
     Quotient,
     Support,
     Support.OrdList,
@@ -55,6 +56,17 @@ executable ons-hs-lstar
     ExampleAutomata,
     IO
 
+executable ons-hs-lstar-perm
+  import: stuff
+  hs-source-dirs: app
+  main-is: LStarPerm.hs
+  build-depends:
+    mtl,
+    ons-hs
+  other-modules:
+    ExampleAutomata,
+    IO
+
 executable ons-hs-teacher
   import: stuff
   hs-source-dirs: app
@@ -97,6 +109,7 @@ test-suite ons-hs-test
   main-is: Spec.hs
   other-modules:
     SpecMap,
+    SpecPermutable,
     SpecSet,
     SpecUtils
   build-depends:
diff --git a/run-lstar-perm.sh b/run-lstar-perm.sh
new file mode 100755
index 0000000..7249cfc
--- /dev/null
+++ b/run-lstar-perm.sh
@@ -0,0 +1,31 @@
+#!/usr/bin/env bash
+
+# Example usage of how to run lstar against a non-interactive teacher. This
+# script will create two fifos for the learner and teacher to communicate over.
+# The communication is not visible, only output to stderr will be shown in
+# the terminal
+
+# safety flags, remove x if you don't like all the output
+set -euxo pipefail
+
+# create temporary directory, and names for the fifo queues (not files)
+tempdir=$(mktemp -d run-lstar.temp.XXXXXX)
+queryfifo="$tempdir/queries"
+answerfifo="$tempdir/answers"
+
+# find the binary for the learner and teacher.
+# The haskell project must be built beforehard (cabal build all)
+lstar=$(cabal list-bin ons-hs-lstar-perm)
+teacher=$(cabal list-bin ons-hs-teacher)
+
+# make the connection for the processes
+mkfifo $queryfifo $answerfifo
+
+# run the teacher in the background
+$teacher < $queryfifo > $answerfifo &
+
+# run the learning algorithm, measuring its time
+time $lstar > $queryfifo < $answerfifo
+
+# clean up
+rm -r $tempdir
diff --git a/src/Permutable.hs b/src/Permutable.hs
new file mode 100644
index 0000000..2488809
--- /dev/null
+++ b/src/Permutable.hs
@@ -0,0 +1,82 @@
+{-# LANGUAGE ImportQualifiedPost #-}
+
+module Permutable where
+
+import Data.List (permutations)
+import Data.Map.Strict qualified as Map
+
+import Nominal
+import Support
+
+---------------------------------
+---------------------------------
+
+-- Invariant: No element occurs more than once
+newtype Perm = Perm (Map.Map Rat Rat)
+  deriving (Eq, Ord, Show)
+
+identity :: Perm
+identity = Perm Map.empty
+
+-- Composition (right to left)
+-- TODO: check this implementation!
+compose :: Perm -> Perm -> Perm
+compose (Perm f) (Perm g) = reduce . Perm $ Map.compose f g <> g <> f
+
+-- Removes elements which are mapped to itself
+reduce :: Perm -> Perm
+reduce (Perm f) = Perm . Map.filterWithKey (\k v -> k /= v) $ f
+
+---------------------------------
+---------------------------------
+
+-- Invariant: The permutation only consists of elements of the support of the
+-- element a.
+-- This is supposed to be a monad. For now, I don't implement the Monad
+-- typeclass, but do everything by hand. (I am not going to use do notation
+-- anyway.)
+data Permuted a = Permuted Perm a
+  deriving (Eq, Ord, Show)
+
+embed :: a -> Permuted a
+embed = Permuted identity
+
+-- to revalidate the invariant
+shrink :: Nominal a => Permuted a -> Permuted a
+shrink (Permuted (Perm m) a) = Permuted (Perm (Map.filter (\p -> elem p (toList (support a))) m)) a
+
+join :: Permuted (Permuted a) -> Permuted a
+join (Permuted f (Permuted g a)) = Permuted (compose f g) a
+
+mapped :: Nominal b => (a -> b) -> Permuted a -> Permuted b
+mapped fun (Permuted f a) = shrink $ Permuted f (fun a)
+
+bind :: Nominal b => (a -> Permuted b) -> Permuted a -> Permuted b
+bind comp (Permuted f a) = case comp a of
+  Permuted g b -> shrink $ Permuted (compose g f) b
+
+allPermutations :: Support -> [Perm]
+allPermutations (Support xs) = fmap (reduce . Perm . Map.fromList . zip xs) . permutations $ xs
+
+-- Returns a lazy list
+allPermuted :: Nominal a => a -> [Permuted a]
+allPermuted el = fmap (flip Permuted el) . allPermutations . support $ el
+
+---------------------------------
+---------------------------------
+
+-- I want Nominal to be a superclass. But for now that gets in the way (as
+-- Permuted is not yet a Nominal type).
+-- Note that acting on an element may change its orbit (as ordered nominal
+-- set).
+class Permutable a where
+  act :: Permuted a -> a
+
+instance Permutable (Permuted a) where
+  act = join
+
+instance Permutable Rat where
+  act (Permuted (Perm m) p) = Map.findWithDefault p p m
+
+instance Permutable a => Permutable [a] where
+  act (Permuted f ls) = fmap (\x -> act (Permuted f x)) ls
diff --git a/test/Spec.hs b/test/Spec.hs
index 94e7069..1990c1f 100644
--- a/test/Spec.hs
+++ b/test/Spec.hs
@@ -10,6 +10,7 @@ import OrbitList (repeatRationals, size)
 import Support (Rat (..))
 
 import SpecMap
+import SpecPermutable
 import SpecSet
 import SpecUtils ()
 
@@ -17,7 +18,7 @@ main :: IO ()
 main = defaultMain allTests
 
 allTests :: TestTree
-allTests = testGroup "main" [setTests, mapTests, countingTests, qcTests]
+allTests = testGroup "main" [setTests, mapTests, countingTests, qcTests, permutableTests]
 
 -- Verifying that the number of orbits is correct. Up to length 7, because
 -- length 8 and longer take at least one second.
diff --git a/test/SpecPermutable.hs b/test/SpecPermutable.hs
new file mode 100644
index 0000000..8a07208
--- /dev/null
+++ b/test/SpecPermutable.hs
@@ -0,0 +1,28 @@
+{-# OPTIONS_GHC -Wno-orphans #-}
+
+module SpecPermutable (permutableTests) where
+
+import Test.Tasty
+import Test.Tasty.HUnit hiding (assert)
+
+import Nominal
+import Permutable
+import Support (Rat (..))
+
+import SpecUtils
+
+permutableTests :: TestTree
+permutableTests = testGroup "Permutable" [assocTest n | n <- [0 .. 6]]
+
+-- For n = 7, this takes roughly 30 seconds!
+assocTest :: Int -> TestTree
+assocTest n =
+  testCase ("associativity " <> show n) $
+    assert and $
+      [lhs f g == rhs f g | f <- perms, g <- perms]
+ where
+  element = fmap (Rat . toRational) $ [1 .. n]
+  supp = support element
+  perms = allPermutations supp
+  lhs f g = act (Permuted (compose f g) element)
+  rhs f g = act (Permuted f (act (Permuted g element)))