[Paper Review] Syntax for Split Preorders

Kosta Došen, Zoran Petrić|arXiv (Cornell University)|Feb 4, 2009

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper presents a syntactic axiomatization of the categories SplPre (split preorders), Gen (split equivalences), and Rel (binary relations) using generators and equations, establishing their algebraic structures—Frobenius for Gen and bialgebra for Rel—within a categorical framework. The key contribution is a completeness proof via normal forms, showing that adding any new equation collapses the system into triviality, thus proving maximality of the presented equational theories.

ABSTRACT

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.

Motivation & Objective

• To provide a syntactic presentation of the category SplPre using generators and equations, capturing split preorders on finite ordinals.
• To show that the subcategory Gen of split equivalences and the category Rel of binary relations admit isomorphic embeddings within SplPre, preserving composition but not identity arrows.
• To reveal the underlying algebraic structures—Frobenius algebra in Gen and bialgebra in Rel—within the framework of SplPre.
• To prove completeness of the syntactic presentations via normal forms, distinguishing SplPre/Gen from the cut-free normal form of Rel.
• To demonstrate that the equational theories of SplPre, Gen, and Rel are maximal: no nontrivial equation can be added without collapsing the system to triviality.

Proposed method

• Define SplPre as the category whose objects are finite ordinals and arrows are split preorders on the disjoint union of source and target sets.
• Present SplPre, Gen, and Rel using a set of generators (e.g., !, ¡, ∇, ∆) and equational axioms, including Frobenius and bialgebra laws.
• Establish isomorphisms between Gen and the category of equivalence relations, and between Rel and a subcategory of SplPre, via structure-preserving maps.
• Construct normal forms for arrow terms in SplPre and Gen, orthogonal to the cut-free normal form of Rel, to prove completeness.
• Use the isomorphism between SplPre and Gen, and between Gen and Rel, to transfer equational reasoning across categories.
• Prove maximality by showing that any non-derivable equation in the respective categories leads to collapse into triviality (e.g., 1₁ = 0₁,₁), implying no further nontrivial equations can be added.

Experimental results

Research questions

• RQ1 How can the category of split preorders on finite ordinals be axiomatized syntactically using generators and equations?
• RQ2 What algebraic structure underlies the category of split equivalences (Gen), and how does it relate to Frobenius algebras?
• RQ3 How is the bialgebraic structure of binary relations (Rel) constructed from the Frobenius structure of Gen within SplPre?
• RQ4 What is the significance of the normal form for SplPre and Gen, and how does it differ from the cut-free normal form of Rel?
• RQ5 Can the equational theories of SplPre, Gen, and Rel be extended with new nontrivial equations without collapsing to triviality?

Key findings

• The syntactic presentation of SplPre, Gen, and Rel is complete, as shown by the existence of normal forms that uniquely represent each arrow term.
• The category Gen carries a Frobenius algebra structure, with the generators !, ¡, ∇, and ∆ satisfying the Frobenius identities.
• The category Rel carries a bialgebraic structure built upon the Frobenius structure of Gen, with the bialgebra laws derived from the composition and duality in SplPre.
• The normal form for SplPre and Gen is orthogonal to the cut-free normal form of Rel, indicating a fundamental syntactic distinction in proof representation.
• Any attempt to add a new equation not already derivable in SplPre, Gen, or Rel leads to triviality—specifically, to the collapse 1₁ = 0₁,₁—proving the maximality of the equational theories.
• The isomorphic image of Rel in SplPre preserves composition but not identity arrows, highlighting a structural distinction between the identity in Rel and the identity in SplPre.