Today I hosted Anna Zamojska-Dzienio from the Warsaw University of Technology. She introduced me to barycentric algebras. Among other things, they allow us to see polytopes and face lattices as the same sort of object ! Intuitively, barycentric algebras are spaces in which one can take the convex combination of elements. I give here my own presentation of what she taught me. Since all of this is new to me there might be errors. Those are mine, not Anna’s.
A barycentric algebra \(\mathcal B\) is an algebra in the sense of “abstract algebra”, consisting of a ground set \(B\), and for each real number \(r\in(0,1)\) a binary operation \(\underline{r}:B\times B\to B\), subject to the following three axioms:
- idempotence : \(\;\underline{r}(a,a)=a\),
- skew-symmetry : \(\;\underline{r}(a,b)=\underline{r}^*(b,a)\), where \(r^*:=1-r\),
- skew-associativity : using \(r\circ s:=(r^*s^*)^*=r+s-rs\) it holds \[\underline r\big(\underline s(a,b),c\big) = \underline{\smash{r\circ s}}\big(a,\underline{\tfrac{r}{r\circ s}}(b,c)\big).\]
The two most instructive examples for barycentric algebras also represent the two extreme cases:
Convex sets. A convex set \(K\subseteq\Bbb R^n\) can be identified as a barycentric algebra by setting
\[\underline{r}(a,b):=(1-r)a+rb.\]
In particular, polytopes are barycentric algebras. In some sense, polytopes are the finitely generated barycentric algebras (more details below).
Semi-lattices. A (join) semi-lattice \(\mathcal L\) with a join operation \(\vee\) is a barycentric algebra under the operation
\[\underline{r}(a,b):=a\vee b.\]
In particular, the face lattice \(\mathcal F(P)\) of a polytope \(P\) is a barycentric algebra.
A barycentric algebra homomorphism is a map \(\phi:\mathcal A\to\mathcal B\) with
\[\phi\big(\underline{r}(a,b)\big)= \underline{r}\big(\phi(a),\phi(b)\big).\]
The relation between polytopes and their face lattices is the following: the map \(P\to\mathcal F(P)\) that sends a point \(x\in P\) to the unique face \(\sigma\in\mathcal F(P)\) that contains \(x\) as an interior point, is a (barycentric algebra) homomorphism. And it is a distinguished homomorphism, as we will see shortly.
The relevant property that separates barycentric algebras of convex sets from the ones definde on semi-lattices turns out to be cancellativity:
\[\underline{r}(a,b)=\underline{r}(a,c) \;\implies\; b=c.\]
Convex sets are cancellative; whereas semi-lattices are “the most non-cancellative” in the sense that they don’t have any non-trivial cancellative sub-algebras. Cancellativity is also a main reason for why we use \(r\in(0,1)\) as opposed to \(r\in[0,1]\): otherwise no barycentric algebra would be cancellative.
Based on this, barycentric algebras are divided into three types:
- geometric type are the cencellative barycentric algebras. They are intended to abstract convex sets.
- combinatorial type have no non-trivial cancellative sub-algebras (in particular, are non-cancellative themselves). They are precisely the semi-lattices.
- mixed type fall into neither of the above classes.
An example of a mixed type is \(\Bbb R\cup\{\infty\}\) with \(\underline{r}(x,\infty)=\infty\) and normal convex combination on all other elements.
There exists a satisfying structure theory for barycentric algebras that separates the geometry from the combinatorics: each barycentric algebra can be subdivided into cancellative sub-algebras (the geometric parts), that when “contracted” leave a semi-lattice (the combinatorial part).
Theorem
For a barycentric algebra \(\mathcal B\) exists a unique homomorphism \(\phi:\mathcal B\to\mathcal L\) onto a semi-lattice \(\mathcal L\) so that all preimages of elements \(\sigma\in\mathcal L\) are cancellative.
For polytopes this distinguished semi-lattice is precisely the face-lattice (minus the empty face).
This provides us with an “abstract algebra” definition of face-lattice! A simplex is a finitely-generated free barycentric algebra; a polytope is a cancellative homomorphic image of a simplex (analogous to how polytopes are projections of simplices); and a face-lattice is a maximal “totally non-cancellative” homomorphic image of a polytope.