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Abstract Barycentric Algebras

Romanowska A.B., Smith J.D.H., Orłowska E.

Fundamenta Informaticae

2007

Vol. 81, nr 1-3

257-273

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 LP-algebra, the algebra of ukasiewicz 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.

The normalization of rectangular algebras

Vanžurová A.

Demonstratio Mathematica

2003

Vol. 36, nr 1

11--28

Our aim is to investigate the normalization (nilpotent shift) (RAr}* of the variety RAr of rectangular algebras of a finite type r. It can be regarded as a common generalization of various algebras studied by different authors, namely of rectangular algebras, [P&R 92, 93], rectangular bands, [Gou], medial grupoids, [J&K], or generalized diagonal algebras, [P 64, 66a, b]. We will determine all subdirectly irreducible algebras in the normalization of the variety of rectangular algebras, give a normal form for terms in the normalization and an algorithm for finding a subdirect decomposition of algberas belonging to the normalization.