An algebraic closure for barycentric algebras and convex sets
Let \(T\) be a subring of the real numbers containing the ring of integers. The open unit interval of \(T\) is then \(I^o(T):=\{r\in T:0<r<1\}\). For each \(p\in I^o(T)\), the authors define a so-called barycentric operation \(\underline{p}:T^2\rightarrow T\) by \(\underline{p}(x,y):=(1-p)x+py\). The members of the variety \(V(T)\) generated by the algebra \((T,\{\underline{p}:p\in I^o(T)\})\) are called (generalized) barycentric algebras. They are basic examples of idempotent and entropic algebras, called modes; see [\textit{A. B. Romanowska} and \textit{J. D. H. Smith}, Modes. Singapore: World Scientific (2002; Zbl 1012.08001)]. The main aim of this paper is to describe some algebraic closure of cancellative barycentric algebras. The latter generalize convex subsets of real affine space \(\mathbb{R}^n\). It is shown that a convex subset of \(\mathbb{R}^n\) is topologically closed iff it is closed in the authors' sense. They consider the following classes: \(Q(T)\), the quasivariety generated by \((T,\{\underline{p}:p\in I^o(T)\})\), \(C(T)\), the quasivariety of cancellative barycentric algebras, and \(H(T)\), some special quasivariety of barycentric algebras. Algebraic closures are constructed here for members of \(H(T)\). Obviously, \(Q(T)\subseteq C(T)\subseteq V(T)\). The main theorem shows that \(Q(T)\subseteq H(T)\subseteq C(T)\) and that each member of \(H(T)\) has a unique closure in \(H(T)\). It also shows when \(H(T)=C(T)\).
scientific article; zbMATH DE number 6110483
scientific article