Definition of the compatibility relation
I have defined a relation $\mathsf{C}$ for sets that captures (to some extent) the notion of compatibility.
In order to do this, we need an operation $': \mathbf{E} \longrightarrow \mathbf{U}$, where $\mathbf{E} \subseteq \mathbf{U}$ and $\mathbf{U}$ is our universe. Intuitively said, the operation $'$ transforms an element of $\mathbf{E}$ into its opposite element. Since it is not necessary that all elements of our domain have an opposite, the domain of $'$ has to be restricted to $\mathbf{E}$, which is the set of oppositable elements of $\mathbf{U}$.
We say that a set $\mathbf{A}$ is compatible with a set $\mathbf{B}$ with respect to an operation $* \in \wp\mathbf{U}^{\wp\mathbf{U}}$, or $\mathsf{C}(\mathbf{A,B})^*$, iff there is no $x\in\mathbf{U}$ such that $x,x' \in (\mathbf{A} \cup \mathbf{B})^*$. We say that $\mathbf{A}$ is incompatible with $\mathbf{B}$ with respect to $*$ iff not $\mathsf{C}(\mathbf{A,B})^*$.
(At this point it may be important to remark that $*$ can be a closure operation. Also remember that $Y^X$ is the set of all functions from $X$ to $Y$ in Halmos notation. Hence, $\wp\mathbf{U}^{\wp\mathbf{U}}$ is the set of all functions from and to sets of $\mathbf{U}$.)
Properties of $\mathsf{C}$
The relation of compatibility $\mathsf{C}$ is then a three-place symmetric relation $\mathsf{C} : \wp\mathbf{U} \times \wp\mathbf{U} \times \wp\mathbf{U}^{\wp\mathbf{U}}$. By symmetric, I mean that $\mathsf{C}(\mathbf{A,B})^* \Leftrightarrow \mathsf{C}(\mathbf{B,A})^*$ holds regardless of the $*$.
But it is no equivalence relation since it is neither reflexive nor transitive. Those properties clearly don't hold when $*$ is increasing and monotone (which is the case for closure operations). In that case, it cannot be reflexive since $\mathsf{C}(\{x,x'\},\{x,x'\})^*$ never holds. Nor can it be transitive because in the hypothesis that $\mathsf{C}(\{x\},\mathbf{B})^*$ and $\mathsf{C}(\mathbf{B},\{x'\})^*$, it still doesn't hold that $\mathsf{C}(\{x\},\{x'\})^*$.
Interpretation
What possible interpretations are there for $\mathbf{U}$, $'$ and $*$ that make sense? The only one I have is in the framework of logic and it goes as follows:
- $\mathbf{U}$: The set of statements or propositions of a formal language ($\mathbf{L}$).
- $'$: The operation of negation ($\neg$) on the statements of $\mathbf{U}$ (or $\mathbf{L}$). (In fact, I conceived $'$ as an extension or generalisation of $\neg$.)
- $*$: A relation of logical consequence ($\vdash: \wp\mathbf{L} \longrightarrow \mathbf{L}$) from sets of statements to statements. (Remember that $p \in \mathbf{A}^\vdash$ iff $\mathbf{A} \vdash p$.)
The question
I have been accepted to present this proposal in the set theory session of a mathematical logic event. In that context, I would like to present some additional interpretations. Regrettably, since my background is mainly philosophical and epistemological, I haven't been able to come up with any other yet. I would grateful to anyone that:
proposes some interesting interpretation(s) for $\mathbf{U}$, $'$ and $*$. Those interpretations can use operations and objects from any field of mathematics (algebra, topology, etc.) or science in general (physics, biology, medicine, etc.). The only requisite is that it makes intuitive sense to use the definition stated above for that interpretation.
refers to research (formal or informal) about a concept similar to compatibility. That research can also come from any branch of science, provided that it could be treated mathematically (or that at least could inspire some mathematical treatment.)
This is my first post in this forum, so let me know if my question is not clear enough and feel free to edit if needed.
Thanks in advance for any feedback. $%**ps:** Might this question be better suited for [Mathematics][1]?$
