Semigroups, ruled continua, and the fixed point property (Q1891647)

\textit{R. J. Koch} and the author [Fundam. Math. 56, 1-8 (1964; Zbl 0126.388)] described a class \({\mathcal C}_1\) of continua called ruled continua. These objects admit the structure of a topological semigroup with a unit and a zero. By a new definition the class \({\mathcal C}_1\) of ruled continua is enlarged in this paper: One adjoins a new class \({\mathcal C}_2\). Then every element of this new class admits a topological semigroup structure with zero and unit (Theorem 2). Let \(X\) be the compact metric continuum constructed by \textit{S. Kinoshita} [Fundam. Math. 40, 96-98 (1953; Zbl 0053.125)]. The author shows that \(X\) is a ruled continuum. It has not the fixed point property. Then two other examples of ruled continua without the fixed point property are given. Finally compact generalized trees are discussed.