Combining entropy structures

Those definitions apply, as we have said, to any subsystem of Cyberspace. We now show how they may be combined and so used to define complicated systems in terms of simpler ones. There are two requirements. We must be able to combine simple views of a given system to endow it with a more realistic complex view; and we must be able to combine different systems to described a larger one equally exactly.

If $\leq_0$ and $\leq_1$ are pre-orders on $X$ then their conjunction is defined to be their intersection

$$x \leq_{\land} x' \ =_{\textit{\tiny def}}\ \ x \leq_0 x' \land x \leq_1 x' .$$

Conjunction corresponds to combining the two orders independently but equally on a given state space. Again, it is clearly a pre-order whose equivalence is the intersection of the two equivalences

$$x~{\sim_{\land}}~x' = x~{\sim}_0~x'\ \land\ x~{\sim}_1~x' .$$

Thus we define the conjunction of two entropy structures $(X,\leq_0,\sim_0)$ and $(X,\leq_1,\sim_1)$ on the same state space to be the entropy structure $(X,\leq_{\land},\sim_{\land})$.

The lexical combination in which pre-order $\leq_0$ is refined by pre-order $\leq_1$ on the same state space, employs (as the name suggests) the latter to refine the former

$$x \leq_{lex} x' \ =_{\textit{\tiny def}}\ \ (x \leq_0 x') \lor (x \sim_0 x' \land x \leq_1 x')$$

where $\sim_0$ denotes the equivalence of $\leq_0$. Investigation of cases shows it to be again a pre-order. Immediately from the definition its equivalence is seen to be

$$x \sim_{lex} x' = x \sim_0 x' \land (x \leq_1 x' \ \lor x'\ \leq_1 x) .$$

Thus, we define the lexical combination of two entropy structures $(X,\leq_0,\sim_0)$ and $(X,\leq_1,\sim_1)$ on the same state space to be the entropy structure $(X,\leq_{lex},\sim_{lex})$.

For pre-orders $(X_0,\leq_0)$ and $(X_1,\leq_1)$ on possibly different state spaces their product is the pre-order $\leq_{\times}$, on the Cartesian product $X_0 \times X_1$ of the state spaces, defined

$$(x_0,x_1) \leq_{\times} (y_0,y_1) \ =_{\textit{\tiny def}}\ \ x_0 \leq_0 y_0 \land x_1 \leq_1 y_1 .$$

It is clearly again a pre-order whose equivalence is, like that for conjunction,

$$(x_0,x_1) \sim_{\times} (y_0,y_1) = x_0 \sim_0 y_0 \land x_1 \sim_1 y_1 .$$

That leads us to define the product of two entropy structures $(X_0,\leq_0,\sim_0)$ and $(X_1,\leq_1,\sim_1)$ to be $(X,\leq_{\times},\sim_{\times})$.

For examples of the way in which a complicated ordering can be obtained on a state space by combining several simple orders using those combinators, see section 5.4.