I was reading "[Origins of the Calculus of Binary Relations](http://boole.stanford.edu/pub/ocbr.pdf)" by Vaughan Pratt where he says "it consists of two components, a logical or static component and a relative or dynamic component" but it seems as if it should be possible to define the "static component" purely in terms of the "dynamic component". Is this the case? 



Calculus of Binary Relations
As I understand it, and in more modern-day terms, the question asks whether it is possible to define the operations $0, 1, \cap, \cup, \neg$ on $P(X^2)$ (operations belonging to the "static component") in terms of "dynamic" operations $\delta', \delta, \circ, \circ', (-)^{op})$ where $\circ$ is relational composition (as an operator on $P(X^2)$):


$$(R \circ S)(x, z) := \exists\_y R(x, y) \wedge S(y, z);$$ 


$\circ'$ is De Morgan dual to $\circ$: 


$$(R \circ' S)(x, z) := \forall\_y R(x, y) \vee S(y, z);$$ 


$\delta \in P(X^2)$ is the diagonal subset (the identity for $\circ$), $\delta'$ is its complement (the identity for $\circ'$), and $(-)^{op}$ is the relational converse: 


$$R^{op}(x, y) = R(y, x).$$ 


**The answer is no**. For example, take $X$ to be $\mathbb{R}$, and consider the relation $n$ where $n(x, y) \Leftrightarrow (x = -y)$. Let $n'$ be the complement of $n$. I claim that the set $A = \{0, 1, \delta, \delta', n, n'\}$ is closed under the dynamic operations. It's clear that each of these elements is a fixed point under relational converse. It's also easy to check that $n \circ n = \delta$, and that 


$$n' \circ n = \delta' = n \circ n',$$ 


$$\delta' \circ n = n' = n \circ \delta',$$ 


$$n' \circ n' = 1 = n' \circ \delta' = \delta' \circ n' = \delta' \circ \delta',$$ 


$$1 \circ x = 1 = x \circ 1 \qquad x \in A, x \neq 0$$ 


$$0 \circ x = 0 = x \circ 0 \qquad x \in A$$


Since $A$ is closed under complementation and under $\circ$, it is also closed under $\circ'$. Thus $A$ is closed under all the dynamic operations. However, it is clearly not closed under intersection, since $n \cap \delta$ is the singleton $\{(0, 0)\}$. 


