lay a significant part in our future interpretation and understanding of this adaptive and self-organizing phenomenon of nature.

***End Tutorial5***

From the examples in Tutorial5, it should be clear that the sequential steps summarized in Fig. \[Fig: Zorn\] are involved in an application of Zorn’s lemma to show that a partially ordered set has a maximal element with respect to its order. Thus for a partially ordered set $(X,\preceq)$, form the set $\mathcal{X}$ of all chains $C$ in $X$. If $C_{+}$ is a maximal chain of $X$ obtained by the Hausdorff Maximal Principle from the chain $\mathcal{C}$ of all chains of $X$, then its supremum $u$ is a maximal element of $(X,\preceq)$. This sequence is now applied, paralleling Example 4.2(1), to the set of arbitrary relations $\textrm{Multi}(X)$ on an infinite set $X$ in order to formulate our definition of chaos that follows.

Let $f$ be a *noninjective map* in $\textrm{Multi}(X)$ and $P(f)$ the number of injective branches of $f$. Denote by $$F=\{ f\in\textrm{Multi}(X)\!:f\textrm{ is a noninjective function on }X\}\subseteq\textrm{Multi}(X)$$ the resulting basic collection of noninjective functions in $\textrm{Multi}(X)$.

\(i) For every $\alpha$ in some directed set $\mathbb{D}$, let $F$ have the extension property $$(\forall f_{\alpha}\in F)(\exists f_{\beta}\in F)\!:P(f_{\alpha})\leq P(f_{\beta})$$

\(ii) Let a partial order $\preceq$ on $\textrm{Multi}(X)$ be defined, for $f_{\alpha},f_{\beta}\in\textrm{Map}(X)\subseteq\textrm{Multi}(X)$ by $$P(f_{\alpha})\leq P(f_{\beta})\Longleftrightarrow f_{\alpha}\preceq f_{\beta},\label{Eqn: chaos1}$$

with $P(f):=1$ for the smallest $f$, define a partially ordered subset $(F,\preceq)$ of $\textrm{Multi}(X)$. This is actually a preorder on $\textrm{Multi}(X)$ in which functions with the same number of injective branches are equivalent to each other.

\(iii) Let $$C_{\nu}=\{ f_{\alpha}\in\textrm{Multi}(X)\!:f_{\alpha}\preceq f_{\nu}\}\in\mathcal{P}(F),\qquad\nu\in\mathbb{D},$$ be the $g$-chains of non-injective functions of