Heaps

Heaps

Idea

A heap is an algebraic structure which is basically equivalent to a group when one forgets about which element is the unit. Similar notions are affine space, principal homogeneous space and so on. However, the notion of a heap has a directness and simplicity in the sense that it is formalized as an algebraic structure with only one ternary operation satisfying a short list of axioms. If we start with a group the ternary operation is defined via $(a,b,c)\mapsto a b^{-1}c$. We can interpret that operation as shifting $a$ by the (right) translation in the group which translates $b$ into $c$. There is also a dual version, quantum heap.

Heaps in the sense of algebra should not be confused with heaps in the sense of theoretical computer science. There are also a number of synonyms for the term ‘heap’; below we consider ‘torsor’ in this light. In Russian one term for a heap is ‘груда’ (‘gruda’) meaning a heap of soil; this is a pun as it is parallel to the russian word ‘группа’ (‘gruppa’) meaning a group: forgetting the unit element is sort of creating an amorphous version. This term also appears in English as ‘groud’.

Definition

A heap $(H,t)$ is a nonempty set $H$ equipped with a ternary operation $t : H \times H \times H\to H$ satisfying the relations

$t(b,b,c) = c = t(c,b,b)$
$t(a,b,t(c,d,e)) = t(t(a,b,c),d,e)$

More generally, a ternary operation in some variety of algebras satisfying the first pair of equations is called a Mal'cev operation. A Mal’cev operation is called associative if it also satisfies the latter equation (i.e. it makes its domain into a heap).

A heap homomorphism, of course, is a function that preserves the ternary operations. This defines a category $Heap$ of heaps.

Automorphism group

As suggested above, if $G$ is a group and we define $t(a,b,c) = a b^{-1} c$, then $G$ becomes a heap. This construction defines a functor $Prin:Grp\to Heap$. In fact, up to isomorphism, all heaps arise in this way; to every heap is associated a group $Aut(H)$ called its automorphism group, unique up to isomorphism. There are a number of ways to define $Aut(H)$ from $H$.

1. If we choose an arbitrary element $e\in H$, then we can define a multiplication on $H$ by $a b = t(a,e,b)$. It is straightforward to verify that this defines a group structure on $H$, whose underlying heap structure is the original one.

2. We can define $Aut(H)$ to be the set of pairs $(a,b)\in H\times H$, modulo the equivalence relation $(a,b)\sim (a',b')$ iff $t(a,a',b')=b$. (We think of $(a,b)$ as representing $a^{-1} b$.) We then define multiplication by $(c,d)(a,b) = (c,t(d,a,b))$; the inverse of (the equivalence class of) $(a,b)$ is (the equivalence class of) $(b,a)$ and the identity element is (the equivalence class of) $(a,a)$ (for any $a$).

3. We can also define $Aut(H)$ as an actual subgroup of the symmetric group of $H$, analogously to Cayley's theorem? (see Wikipedia) for groups. We take the elements of $Aut(H)$ to be set bijections of the form $t(-,a,b): H \rightarrow H$ where $a,b \in H$, with composition as the group operation. Note that

$t(-,c,d) \cdot_{Aut(H)} t(-, a,b) = t(t(-,c,d),a,b) = t(-,c,t(d,a,b)),$

so $Aut(H)$ is closed under this operation. The first axiom of a heap shows that $Aut(H)$ contains the identity $t(-,x,x)$ for any $x$), and the inverse of $t(\cdot,a,b)$ is $t(\cdot,b,a)$; thus $Aut(H)$ is a subgroup of the symmetric group of $H$.

Note that in both the second and third constructions, the elements of $Aut(H)$ are determined by pairs of elements of $H$, modulo some equivalence relation. The following theorem shows that the two equivalence relations are the same.

Theorem

The following are equivalent

1. bijections $t(\cdot,a,b)$ and $t(\cdot,a',b')$ are the same maps,

2. $t(a,a',b') = b$,

3. $t(b,b',a') = a$.

Proof

(ii) follows from (i) and $t(a,a,b) = b$.

(iii) follows from (ii) by applying $t(\cdot,b',a')$ on the right. Similarly (ii) follows from (iii).

(i) follows from (ii) by the calculation:

$t(x,a',b') = t(t(x,a,a),a',b')= t(x,a,t(a,a',b')) = t(x,a,b).$

The composition laws are also easily seen to agree, so the second two constructions of $Aut(H)$ are canonically isomorphic. To compare them to the first construction, observe that for a fixed $e\in H$, any equivalence class contains a unique pair of the form $(e,a)$. (If $(b,c)$ is in the equivalence class, then $a$ is determined by $a = t(e,b,c)$.) This sets up a bijection between the first two constructions, which we can easily show is an isomorphism.

The second two constructions are clearly functorial, so we have a functor $Aut:Heap\to Grp$. Note that we have $Aut(Prin(G))\cong G$ for any group $G$, and $Prin(Aut(H))\cong H$ for any heap $H$, but while the first isomorphism is natural, the second is not. In particular, the categories $Heap$ and $Grp$ are not equivalent.

Heaps and torsors

Note that $Aut(H)$ comes equipped with a canonical action on $H$ (this is most clear from the third definition). This action is transitive (by $t(a,a,b) = b$) and free (if $t(a,b,c) = a$ then by the previous statement $t(x,b,c) = x$ for each $x$, and in particular $t(b,b,c) = b$ and also $t(b,b,c) = c$). Therefore, $H$ is an $Aut(H)$-torsor (over a point). Conversely, a torsor $H$ over any group $G$ can be made into a heap, by defining $t(a,b,c) = g\cdot c$, where $g\in G$ is the unique group element such that $g\cdot b = a$.

In fact, the category $Heap$ is equivalent to the following category $Tors$: its objects are pairs $(G,H)$ consisting of a group $G$ and a $G$-torsor $H$, and its morphisms are pairs $(\phi,f):(G,H)\to (G',H')$ consisting of a group homomorphism $\phi:G\to G'$ and a $\phi$-equivariant map $f:H\to H'$.

The empty heap

If we wish $Heap$ to be an algebraic category, then we must remove the clause that the underlying set of a heap must be nonempty. Then the empty set becomes a heap in a unique way. However, in this case, the various theorems relating heaps to groups above all break down. For this reason, one usually requires a heap to be inhabited.

On the other hand, we could generalize the notion of group to allow for an empty group. This even remains a purely algebraic notion: we can define a group as a (traditionally nonempty) set equipped with a binary operation (to be thought of as $a, b \mapsto a/b \coloneqq a b^{-1}$) satisfying these laws:

• $a/a = b/b$,
• $(a/a)/((a/a)/a) = a$,
• $a/(b/c) = (a/((c/c)/c))/b$.

Then any possibly-empty-group is a possibly-empty-heap, and every possibly-empty-heap arises in this way from its automorphism possibly-empty-group (defined by either method (2) or (3)); the category of possibly-empty-heaps is equivalent to the category of possibly-empty-groups equipped with torsors over the point; etc.

This is even constructive; the theorems can be proved uniformly, rather than by treating the empty and inhabited cases separately. (This rather trivial method is obvious to a classical mathematician, but it's not constructively valid, since a possibly-empty-group/heap as defined here can't be constructively proved empty or inhabited; it can only be proved empty iff not inhabited. Indeed, taking any group $G$ and any truth value $P$, the possibly-empty-subgroup $\{x \in G \;|\; P\}$ is empty or inhabited iff $P$ is false or true.)

References and remarks

• G.M. Bergman, A.O. Hausknecht, Cogroups and co-rings in categories of associative rings, Ch.IV, paragraph 22, p.95ff – Providence, R.I. : AMS 1996.

• Z. Škoda, Quantum heaps, cops and heapy categories, Mathematical Communications 12, No. 1, pp. 1–9 (2007); (math.QA/0701749)

• wikipedia:heap

• Heaps and torsors

There is an oidification (horizontal categorification) of a heap, sometimes called a heapoid.

Last revised on January 3, 2018 at 16:32:17. See the history of this page for a list of all contributions to it.