# Anna Zamojska-Dzienio

## Barycentric algebras – convexity and order

Originating from the use of barycentric coordinates in geometry by Möbius (1827), barycentric algebras were introduced in the nineteen-forties for the axiomatization of real convex sets, presented algebraically with binary operations given by weighted means, the weights taken from the (open) unit interval in the real numbers.

Barycentric algebras unify ideas of convexity and order. They provide intrinsic descriptions of convex sets (cancellative b.a.), semilattices (commutative and associative b.a. with all operations being equal), and more general semilattice-ordered systems of convex sets, independently of any ambient affine or vector space. Natural applications can be found in physics, biology, and social sciences for the modeling of systems that function on (potentially incomparable) multiple levels, and also in computational geometry to analyze systems of barycentric coordinates.

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.

Some references:

- A.F. Möbius, Der Baryzentrische Calcul, Barth, Leipzig, 1827. Available at https://doi.org/10.3931/e-rara-14538
- A.B. Romanowska, J.D.H. Smith, Modes, World Scientific, 2002.
- J.D.H. Smith, On the Mathematical Modeling of Complex Systems, Center for Advanced Studies, Warsaw University of Technology, Warsaw, 2013.
- A.B. Romanowska, J.D.H. Smith, A. Zamojska-Dzienio, Barycentric algebra and convex polygon coordinates. Preprint: arXiv: 2308.11634
- M.H. Stone, Postulates for the barycentric calculus, Annali di Matematica Pura ed Applicada 29 (1949), 25–30.