Barycentric algebras -- convexity and order

This is the abstract of a series of lectures given during the XIIIth School on Geometry and Physics, Bialystok (Poland), in July 2024. In this minicourse, we first examine the algebraic aspects of barycentric algebras. Then, we focus on various examples and applications, reviewing the pertinence of the barycentric algebra structure.

💡 Research Summary

The paper presents a comprehensive study of barycentric algebras, a class of algebraic structures that abstract the weighted‑mean operations (1‑p)·x + p·y with parameters p taken from the open unit interval I◦ = (0, 1). By fixing a subfield R ⊂ ℝ and restricting the parameters to I◦ ∩ R, the authors obtain a variety B of barycentric algebras that simultaneously captures the notions of convexity and order.

The exposition begins with a reformulation of affine spaces as algebras (A,R) whose basic binary operations are indexed by all real numbers. Affine subspaces, affine maps, and products are then described as subalgebras, homomorphisms, and Cartesian products, respectively. This viewpoint sets the stage for treating convex subsets of ℝⁿ as algebras (C,I◦) where the operations are indexed only by the open interval. The authors emphasize that using the open interval rather than the closed one is essential for preserving regularity of the defining identities; inclusion of the projection operations 0 and 1 would break this regularity and prevent the development of the subsequent structure theory.

Barycentric algebras are axiomatized by three regular identities: (a) idempotence p(x,x)=x, (b) skew‑commutativity p(x,y)=1−p(y,x), and (c) skew‑associativity p(r(x,y),z)=r∘p(x,p/(r∘p)(y,z)) where r∘p=r+p−rp. These identities hold uniformly for every choice of parameters p,r∈I◦, guaranteeing that the variety B is equationally defined and closed under subalgebras, products, and homomorphic images.

A series of examples illustrates the breadth of the theory. The standard example (V,I◦) with V a real vector space recovers the usual convex subsets as subalgebras. Semilattices become barycentric algebras when each p‑operation is identified with the join operation ∨; this yields the iterated semilattice subvariety S of B, characterized by the identity p(x,y)=r(x,y) for all p,r. The authors also give a counter‑example showing that a homomorphic image of a convex set need not be convex, thereby demonstrating that the class C of convex sets is not a variety, while its homomorphic closure B is.

The core structural results are two decomposition theorems. The first (Theorem 5.1) shows that every barycentric algebra A admits a largest semilattice quotient π:A→S, called the semilattice replica. For each s∈S, the fiber π⁻¹({s}) is an open convex subalgebra, and A is the disjoint union of these fibers—hence a semilattice sum of open convex sets. The notion of a “wall” is introduced to describe subalgebras that are closed under all p‑operations; open barycentric algebras have no non‑empty walls.

The second theorem (Theorem 5.5) refines the decomposition using Plonka sums. Viewing the semilattice replica S as a small category, a functor F:S→C assigns to each s a convex set C_s and to each morphism s≤t a homomorphism φ_{s,t}. The Plonka sum of the family (C_s)_{s∈S} reconstructs a barycentric algebra that contains A as a subalgebra. This construction generalizes the classical Plonka sum of semigroups to arbitrary algebras and provides a systematic method to rebuild any barycentric algebra from its convex fibers and the ordering encoded in S.

The paper then applies this framework to convex polytopes. Vertices, edges, and higher‑dimensional faces correspond to walls and to the convex fibers in the semilattice decomposition. Consequently, a polytope is precisely a semilattice sum of the interiors of its faces, and its Plonka representation is obtained by taking the faces themselves as the convex components with inclusion maps as the connecting homomorphisms.

Finally, the authors discuss the relevance of barycentric algebras to complex systems. Because a barycentric algebra simultaneously encodes a family of convex sets together with a partial order (the semilattice replica), it serves as a natural algebraic model for hierarchical biological networks, multi‑level statistical mechanics, non‑deterministic and probabilistic computational systems, and thermodynamic state spaces. The paper concludes by outlining future research directions, including categorical extensions, quantitative analysis of the semilattice replica, and applications to optimization and game theory.