To motivate the question, consider the theory of rings. Define $x \parallel y$ to mean $\exists w \exists z .((x - y) z = w (x - y) = 1)$, or in words, "$x - y$ is a unit". Then $\parallel$ is a binary relation with the following properties:

• Naturality. For all homomorphisms $\phi$, $x \parallel y$ implies $\phi (x) \parallel \phi (y)$.

• Algebraic irreflexivity. $=$ and $\parallel$ are algebraically inconsistent, in the sense that: $$\forall x \forall y \forall z . (x \parallel x \text{ implies } y = z)$$

A natural algebraic irreflexive relation (for a one-sorted theory) is a binary relation with the two formal properties above. The word "algebraic" has a double meaning here: attached to "natural", it signals that we are talking about preservation by algebra homomorphisms rather than, say, elementary embeddings; and attached to "irreflexive", it signals that we are weakening the definition in order to accommodate algebraic examples.

Actually, we can generalise to any category $\mathcal{A}$ equipped with a functor $U : \mathcal{A} \to \textbf{Set}$: we are then talking about a functor $[{\parallel}] : \mathcal{A} \to \textbf{Set}$ equipped with a natural monomorphism $[{\parallel}] \hookrightarrow U \times U$ such that $$([{=}] \cap [{\parallel}]) \times U \times U \subseteq [{=}] \times [{=}]$$ as subfunctors of $U \times U \times U \times U$, where $[{=}]$ is the diagonal of $U \times U$. Furthermore, let us say $\parallel$ is representable (resp. weakly representable, multirepresentable, etc.) if the functor $[{\parallel}]$ is representable (resp. weakly representable, multirepresentable, etc.). The above example in the theory of rings is representable. In the logical picture, this is closely related to definability of $\parallel$ by formulae in various complexity classes.

Question. Are there any other interesting examples of natural algebraic irreflexive relations? Particularly, are there examples in algebraic theories that are not ring-like? (Here, by "ring-like" I mean having two unital binary operations where the unit for one of the binary operations is absorbing for the other.)

Note that there is a class of trivial examples obtained by defining $\parallel$ to be the empty relation. Another class of uninteresting examples is obtained when $U : \mathcal{A} \to \textbf{Set}$ factors through the subcategory of injective maps – in that case, the denial inequality $\ne$ has the naturality property. One way of getting such a category is to take a relational theory that already has an irreflexive relation – such as the theory of strict partial orders – but this too is not very interesting (other than for demonstrating that symmetry is not automatic).

Here is a non-trivial example, but still in the ring-like realm. In the theory of rigs (i.e. rings without negatives, also called semirings), define $x \parallel y$ to mean: $$\exists s \exists t \exists u \exists v. (s x + t y = x u + y v = 1 \text{ and } x y = 0)$$ Then $\parallel$ is a natural algebraic irreflexive relation and is weakly representable in the sense that there is a representable functor $C$ and a natural transformation $C \Rightarrow U \times U$ whose image is the subfunctor $[{\parallel}]$.

The failure of representability in the above is essentially because the witnesses $s, t, u, v$ are not necessarily unique. In the theory of (bounded but not necessarily distributive) lattices, we can analogously define $x \parallel y$ by $$x \lor y = \top \text{ and } x \land y = \bot$$ to obtain a representable natural algebraic irreflexive relation.