An algebraic closure for barycentric algebras and convex sets

作者：Czedli Gabor^*; Romanowska A B

来源：Algebra Universalis, 2012, 68(1-2): 111-143.

DOI：10.1007/s00012-012-0195-y

• mode
• barycentric algebra
• medial groupoid
• entropic groupoid
• entropic algebra
• convex set
• algebraic closure
• closed algebra

Let A be an algebra (of an arbitrary finitary type), and let gamma be a binary term. A pair (a, b) of elements of A will be called a gamma-eligible pair if for each x in the subalgebra generated by {a, b} such that x is distinct from a there exists an element y in A such that b = xy gamma. We say that A is a gamma-closed algebra if for each gamma-eligible pair (a, b) there is an element c with b = ac gamma. We call A a closed algebra if it is gamma-closed for all binary terms gamma that do not induce a projection. %26lt;br%26gt;Let T be a unital subring of the field of real numbers. Equipped with all the binary operations (x, y) bar right arrow (1 - p) x + py for p is an element of T and 0 %26lt; p %26lt; 1, T becomes a mode, that is, an idempotent algebra in which any two term functions commute. In fact, the mode T is a (generalized) barycentric algebra. Let Q(T) denote the quasivariety generated by this mode. %26lt;br%26gt;Our main theorem asserts that each mode of Q(T) extends to a minimal closed cancellative mode, which is unique in a reasonable sense. In fact, we prove a slightly stronger statement. As corollaries, we obtain a purely algebraic description of the usual topological closure of convex sets, and we exemplify how to use the main theorem to show that certain open convex sets are not isomorphic.

• 出版日期2012-10