Abstract Barycentric Algebras

Issue title: Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science. In Recognition of Professor Andrzej Grzegorczyk

Article type: Research Article

Authors: Romanowska, Anna B. | Smith, Jonathan D.H. | Orłowska, Ewa

Affiliations: Faculty of Mathematics and Information Sc., Warsaw University of Technology, 00-661 Warsaw, Poland. E-mail: aroman@alpha.mini.pw.edu.pl | Department of Mathematics, Iowa State University Ames, Iowa, 50011, USA. E-mail: jdhsmith@math.iastate.edu | National Institute of Telecommunications, 04 854 Warsaw, Poland. E-mail: E.Orlowska@itl.waw.pl

Abstract: This paper presents a new approach to the study of (real) barycentric algebras, in particular convex subsets of real affine spaces. Barycentric algebras are cast in the setting of two-sorted algebras. The real unit interval indexing the set of basic operations of a barycentric algebra is replaced by an LΠ-algebra, the algebra of Łukasiewicz Product Logic. This allows one to define barycentric algebras abstractly, independently of the choice of the unit real interval. It reveals an unexpected connection between barycentric algebras and (fuzzy) logic. The new class of abstract barycentric algebras incorporates barycentric algebras over any linearly ordered field, the B-sets of G. M. Bergman, and E. G. Manes' if-then-else algebras over Boolean algebras.

Keywords: affine space, barycentric algebra, mode, cancellative mode, LΠ-algebra, Łukasiewicz Product Logic, heterogeneous algebra, B-set, if-then-else algebra, rectangular algebra

Journal: Fundamenta Informaticae, vol. 81, no. 1-3, pp. 257-273, 2007