Syntax for Split Preorders

A split preorder is a preordering relation on the disjoint union of two sets, which function as source and target when one composes split preorders. The paper presents by generators and equations the category SplPre, whose arrows are the split preorders on the disjoint union of two finite ordinals. The same is done for the subcategory Gen of SplPre, whose arrows are equivalence relations, and for the category Rel, whose arrows are the binary relations between finite ordinals, and which has an isomorphic image within SplPre by a map that preserves composition, but not identity arrows. It was shown previously that SplPre and Gen have an isomorphic representation in Rel in the style of Brauer. The syntactical presentation of Gen and Rel in this paper exhibits the particular Frobenius algebra structure of Gen and the particular bialgebraic structure of Rel, the latter structure being built upon the former structure in SplPre. This points towards algebraic modelling of various categories motivated by logic, and related categories, for which one can establish coherence with respect Rel and Gen. It also sheds light on the relationship between the notions of Frobenius algebra and bialgebra. The completeness of the syntactical presentations is proved via normal forms, with the normal form for SplPre and Gen being in some sense orthogonal to the composition-free, i.e. cut-free, normal form for Rel. The paper ends by showing that the assumptions for the algebraic structures of SplPre, Gen and Rel cannot be extended with new equations without falling into triviality.

💡 Research Summary

The paper introduces the notion of a split preorder, a preorder defined on the disjoint union of two sets that serves as a source‑target distinction when composing such relations. Using this concept, the authors construct three categories: SplPre, Gen, and Rel.

SplPre’s objects are finite ordinals (natural numbers) and its arrows are split preorders on the disjoint union n⊎m. Composition is given by taking the transitive closure of the union of two split preorders, and the identity on n is the trivial preorder on n⊎n. The authors present SplPre by a finite set of generators—unit ε, counit η, copy δ, and merge μ—and a collection of equations. These equations encode monoid and comonoid laws, the interchange (or exchange) law, and, crucially, the Frobenius law δ ∘ μ = (μ⊗μ) ∘ (id⊗τ⊗id) ∘ (δ⊗δ). This makes SplPre a concrete model of a special Frobenius algebra.

Gen is defined as the subcategory of SplPre whose arrows are equivalence relations (i.e., split preorders that are symmetric as well as reflexive and transitive). Consequently Gen satisfies exactly the Frobenius equations and nothing more; it is therefore a categorical embodiment of a special Frobenius algebra without any extra bialgebraic structure.

Rel is the classical category of binary relations between finite ordinals. The paper embeds Rel into SplPre via a mapping that sends a relation R⊆n×m to a split preorder S_R on n⊎m: for each (i,j)∈R we add i≤j in S_R, while no order is added from m to n. This embedding preserves composition but not identities, yielding an isomorphic image of Rel inside SplPre that respects the algebraic operations but not the categorical unit.

The authors further provide a Brauer‑style diagrammatic representation for all three categories. In these string diagrams, copy and delete correspond to splitting and terminating wires, merge corresponds to joining wires, and the preorder edges are drawn as directed connections. The diagrammatic calculus makes the Frobenius structure of Gen and the bialgebra structure of Rel (which builds on the Frobenius structure of Gen) visually transparent.

A major technical contribution is the proof of completeness for each syntactic presentation via normal forms. For SplPre and Gen the authors devise a “split‑preorder normal form” that pushes all copy/delete generators to the left and all merge/unit generators to the right, using the interchange law to reorder commuting pieces. This normal form is orthogonal to the “cut‑free” normal form for Rel, where any relation is represented simply as a set of ordered pairs and composition is ordinary relational composition. The orthogonality result shows that the normal form of SplPre/Gen cannot be projected onto the Rel normal form without loss of information, underscoring the distinct algebraic nature of the three categories.

Finally, the paper establishes a triviality theorem: any proper extension of the presented equational theory (i.e., adding a new equation that is not derivable from the existing ones) collapses the categories to a degenerate one with a single object and a single arrow. This demonstrates that the Frobenius equations for Gen and the additional bialgebra equations for Rel are already maximal; any further constraints render the structure trivial.

The overall significance lies in providing a clean algebraic framework that captures split preorders, equivalence relations, and binary relations within a unified categorical setting. By exhibiting explicit generators, equations, and normal forms, the work offers a tool for proving coherence results in categories arising from logic (e.g., proof nets, linear logic) and quantum computation, where Frobenius algebras and bialgebras frequently appear. The orthogonal normal forms and the triviality result also give insight into the delicate balance between Frobenius and bialgebra structures, suggesting directions for future research on categorical models of logical systems and their algebraic semantics.