order preserving mapping

For each point p of a continuum X equipped with an arc-structure $A : X \times X \to C(X)$ we define a partial order $\le_p$ by letting $x \le_p y$ whenever $A(p,x) \subset A(p,y)$. Let X and Y be continua with fixed arc-structures A and B, respectively. We say that a surjective mapping $f: X \to Y$ is a $\le_p$-mapping provided that $x \le_p y$ in X implies that $f(x) \le_{f(p)} f(y)$ in Y. If, in addition, $Y \subset X$, B = A|(Y x Y), and f is a retraction, then f is called a $\le_p$-retraction (or $\le_p$- preserving retraction). The concept of a <_p-mapping is defined in a similar manner (with $f(x) \ne f(y)$ implied by $x \ne y$). For order preserving mappings see e.g. [42, I.7, p. 553].