diff --git a/algebra/Z2.h b/algebra/Z2.h
index ea4514b..f13f192 100644
--- a/algebra/Z2.h
+++ b/algebra/Z2.h
@@ -13,6 +13,7 @@ class Z2
   Z2 (int x) : v((bool)(x & 1)) { }
   Z2 (bool v_) : v(v_) { }
   Z2 (const Z2 &x) : v(x.v) { }
+  Z2 (copy, const Z2 &x) : v(x.v) { }
   ~Z2 () { }
   
   Z2 &operator = (const Z2 &x) { v = x.v; return *this; }
diff --git a/algebra/multivariate_laurentpoly.h b/algebra/multivariate_laurentpoly.h
index cf4df00..49c2137 100644
--- a/algebra/multivariate_laurentpoly.h
+++ b/algebra/multivariate_laurentpoly.h
@@ -56,15 +56,6 @@ class multivariate_laurent_monomial
     return m == e.m;
   }
   
-  bool operator < (const multivariate_laurent_monomial &e) const
-  {
-#ifndef NDEBUG
-    check ();
-    e.check ();
-#endif
-    return m < e.m;
-  }
-  
   bool operator == (int x) const
   {
     assert (x == 1);
@@ -76,6 +67,24 @@ class multivariate_laurent_monomial
   
   bool operator != (int x) const { return !operator == (x); }
   
+  bool operator < (const multivariate_laurent_monomial &e) const
+  {
+#ifndef NDEBUG
+    check ();
+    e.check ();
+#endif
+    return m < e.m;
+  }
+  
+  bool operator <= (const multivariate_laurent_monomial &e) const
+  {
+#ifndef NDEBUG
+    check ();
+    e.check ();
+#endif
+    return m <= e.m;
+  }
+  
   multivariate_laurent_monomial &operator *= (const multivariate_laurent_monomial &e)
   {
     for (map<unsigned, int>::const_iter i = e.m; i; i ++)
@@ -203,7 +212,7 @@ class multivariate_laurentpoly
   
   multivariate_laurentpoly (const multivariate_laurentpoly &p) : coeffs(p.coeffs) { }
   multivariate_laurentpoly (copy, const multivariate_laurentpoly &p)
-    : coeffs(COPY, p.coeffs)
+    : coeffs(COPY2, p.coeffs)
   {
   }
 
@@ -226,7 +235,7 @@ class multivariate_laurentpoly
     return *this;
   }
   
-  bool operator == (multivariate_laurentpoly p) const
+  bool operator == (const multivariate_laurentpoly &p) const
   {
 #ifndef NDEBUG
     check ();
@@ -235,9 +244,6 @@ class multivariate_laurentpoly
     return coeffs == p.coeffs;
   }
   
-  unsigned card () const { return coeffs.card (); }
-  pair<monomial, T> head () const { return coeffs.head (); }
-  
   bool operator == (int x) const
   {
 #ifndef NDEBUG
@@ -259,9 +265,30 @@ class multivariate_laurentpoly
   
   bool operator != (int x) const { return !operator == (x); }
   
-  multivariate_laurentpoly &operator += (multivariate_laurentpoly p);
-  multivariate_laurentpoly &operator -= (multivariate_laurentpoly p);
-  multivariate_laurentpoly &operator *= (multivariate_laurentpoly p)
+  bool operator < (const multivariate_laurentpoly &p) const
+  {
+#ifndef NDEBUG
+    check ();
+    p.check ();
+#endif
+    return coeffs < p.coeffs;
+  }
+  
+  bool operator <= (const multivariate_laurentpoly &p) const
+  {
+#ifndef NDEBUG
+    check ();
+    p.check ();
+#endif
+    return coeffs <= p.coeffs;
+  }
+  
+  unsigned card () const { return coeffs.card (); }
+  pair<monomial, T> head () const { return coeffs.head (); }
+  
+  multivariate_laurentpoly &operator += (const multivariate_laurentpoly &p);
+  multivariate_laurentpoly &operator -= (const multivariate_laurentpoly &p);
+  multivariate_laurentpoly &operator *= (const multivariate_laurentpoly &p)
   {
     return operator = (*this * p);
   }
@@ -290,14 +317,14 @@ class multivariate_laurentpoly
   multivariate_laurentpoly &muladdeq (multivariate_laurentpoly a, multivariate_laurentpoly b);
   
   multivariate_laurentpoly operator - () const { return multivariate_laurentpoly () - *this; }
-  multivariate_laurentpoly operator + (multivariate_laurentpoly p) const
+  multivariate_laurentpoly operator + (const multivariate_laurentpoly &p) const
   {
     multivariate_laurentpoly r (COPY, *this);
     r += p;
     return r;
   }
   
-  multivariate_laurentpoly operator - (multivariate_laurentpoly p) const
+  multivariate_laurentpoly operator - (const multivariate_laurentpoly &p) const
   {
     multivariate_laurentpoly r (COPY, *this);
     r -= p;
@@ -330,7 +357,7 @@ operator * (const T &s, const multivariate_laurentpoly<T> &p)
 }
 
 template<class T> multivariate_laurentpoly<T> &
-multivariate_laurentpoly<T>::operator += (multivariate_laurentpoly p)
+multivariate_laurentpoly<T>::operator += (const multivariate_laurentpoly &p)
 {
   for (typename map<monomial, T>::const_iter i = p.coeffs; i; i ++)
     {
@@ -344,7 +371,7 @@ multivariate_laurentpoly<T>::operator += (multivariate_laurentpoly p)
 }
 
 template<class T> multivariate_laurentpoly<T> &
-multivariate_laurentpoly<T>::operator -= (multivariate_laurentpoly p)
+multivariate_laurentpoly<T>::operator -= (const multivariate_laurentpoly &p)
 {
   for (typename map<monomial, T>::const_iter i = p.coeffs; i; i ++)
     {
diff --git a/lib/io.cpp b/lib/io.cpp
index c900a73..a02ccd0 100644
--- a/lib/io.cpp
+++ b/lib/io.cpp
@@ -157,3 +157,33 @@ reader::read_uint64 ()
     }
   return x;
 }
+
+void
+read (reader &r, std::string &s)
+{
+  unsigned n = r.read_unsigned ();
+  
+  char buf[n + 1];
+  unsigned k = fread (buf, sizeof (char), n + 1, r.fp);
+  if (k != n + 1)
+    {
+      stderror ("fread");
+      exit (EXIT_FAILURE);
+    }
+  assert (buf[n] == 0);
+  
+  s = std::string (buf);
+}
+
+void
+write (writer &w, const std::string &s)
+{
+  unsigned n = s.length ();
+  w.write_unsigned (n);
+  
+  if (fwrite (s.c_str (), sizeof (char), n + 1, w.fp) != n + 1)
+    {
+      stderror ("fwrite");
+      exit (EXIT_FAILURE);
+    }
+}
diff --git a/lib/io.h b/lib/io.h
index 5ce6f9c..9f499e9 100644
--- a/lib/io.h
+++ b/lib/io.h
@@ -45,6 +45,7 @@ inline void read (reader &r, char &x) { x = r.read_char (); }
 inline void read (reader &r, int &x) { x = r.read_int (); }
 inline void read (reader &r, unsigned &x) { x = r.read_unsigned (); }
 inline void read (reader &r, uint64 &x) { x = r.read_uint64 (); }
+void read (reader &r, std::string &s);
 template<class T> inline void read (reader &r, T &x) { x = T(r); }
 
 inline void ctor_read (reader &r, bool *p) { *p = r.read_bool (); }
@@ -59,4 +60,5 @@ inline void write (writer &w, char x) { w.write_char (x); }
 inline void write (writer &w, int x) { w.write_int (x); }
 inline void write (writer &w, unsigned x) { w.write_unsigned (x); }
 inline void write (writer &w, uint64 x) { w.write_uint64 (x); }
+void write (writer &w, const std::string &s);
 template<class T> inline void write (writer &w, const T &x) { x.write_self (w); }
diff --git a/lib/map.h b/lib/map.h
index cb3ea78..3be8881 100644
--- a/lib/map.h
+++ b/lib/map.h
@@ -11,6 +11,7 @@ class map : public map_wrapper<std::map<K, V>, K, V>
   map (unsigned dummy_size) : base(dummy_size) { }
   map (const map &m) : base(m) { }
   map (copy, const map &m) : base(COPY, m) { }
+  map (copy2, const map &m) : base(COPY2, m) { }
   map (reader &r) : base(r) { }
   ~map () { }
   
diff --git a/lib/map_wrapper.h b/lib/map_wrapper.h
index 8aae708..762bfc7 100644
--- a/lib/map_wrapper.h
+++ b/lib/map_wrapper.h
@@ -28,6 +28,7 @@ class map_wrapper
   map_wrapper (unsigned dummy_size) : impl(new map_impl) { }
   map_wrapper (const map_wrapper &m) : impl(m.impl) { }
   map_wrapper (copy, const map_wrapper &m) : impl(new map_impl) { impl->t = M (m.impl->t); }
+  map_wrapper (copy2, const map_wrapper &m);
   map_wrapper (reader &r);
   ~map_wrapper () { }
   
@@ -134,6 +135,15 @@ class map_wrapper
   void write_self (writer &w) const;
 };
 
+template<class M, class K, class V>
+map_wrapper<M, K, V>::map_wrapper (copy2, const map_wrapper &m)
+  : impl(new map_impl)
+{
+  /* Keys are immutable.  Just copy the values. */
+  for (const_iter i = m; i; i ++)
+    push (i.key (), V (COPY, i.val ()));
+}
+
 template<class M, class K, class V>
 map_wrapper<M, K, V>::map_wrapper (reader &r)
   : impl(new map_impl)
diff --git a/main.cpp b/main.cpp
index 5c7a627..074fc13 100644
--- a/main.cpp
+++ b/main.cpp
@@ -143,23 +143,50 @@ main ()
   }
 #endif
   
-  map<std::string, int> m;
-  m.push ("foo", 3);
-  m.push ("barz", 4);
-  m.push ("pazazz", 39);
+  multivariate_laurentpoly<Z> p = -11;
+  p.muladdeq (5, VARIABLE, 1);
+  p.muladdeq (7, VARIABLE, 2);
+  p.muladdeq (-3, VARIABLE, 3);
+
+  multivariate_laurentpoly<Z> q = p*p + p + 23;
+  multivariate_laurentpoly<Z> r = q*q - Z (7)*p + 81;
   
-  assert (m % "foo");
-  assert (m("foo") == 3);
-  assert (! (m % "fop"));
+  multivariate_laurentpoly<Z> s = r - p*q + 10;
+
+  display ("p:", p);
+  display ("q:", q);
+  display ("r:", r);
+  display ("s:", s);
   
-  hashmap<std::string, int> m2;
-  m2.push ("foo", 3);
-  m2.push ("barz", 4);
-  m2.push ("pazazz", 39);
+  map<multivariate_laurentpoly<Z>, std::string> m;
+  m.push (p, "p");
+  m.push (q, "q");
+  m.push (r, "thisisr");
   
-  assert (m2 % "foo");
-  assert (m2("foo") == 3);
-  assert (! (m2 % "fop"));
+  assert (m % p);
+  assert (m % q);
+  assert (m % r);
+  assert (! (m % s));
+  
+  assert (m(p) == "p");
+  assert (m(q) == "q");
+  assert (m(r) == "thisisr");
+  
+  std::string str ("This is a test.");
+  
+  {
+    writer w ("test.dat");
+    write (w, m);
+  }
+  
+  reader rdr ("test.dat");
+  map<multivariate_laurentpoly<Z>, std::string> m2 (rdr);
+    
+  assert (m == m2);
+    
+  assert (m2(p) == "p");
+  assert (m2(q) == "q");
+  assert (m2(r) == "thisisr");
   
 #if 0
   test_ring<Z2> (2);
diff --git a/todo.txt b/todo.txt
index f043efd..ad38695 100644
--- a/todo.txt
+++ b/todo.txt
@@ -9,6 +9,8 @@ in knotkit/
  - unify chain_complex_simplifer and build_sseq
  - remove hardcoded limits (max_...)
  - revive testlib
+ - figure out proper copy interface and consistently support copy and
+   deepcopy
 
 in lib/
  - add hashset