Arrow-Type Semigroupoids Overview

• Arrow-type semigroupoids are algebraic structures with partially defined associative compositions that enforce strict typing of arrows.
• They generalize semigroups and categories, offering a framework to model typed computational processes and transformation systems.
• Their study aids in automata theory, operator algebras, and categorical algebra by providing canonical invariants of type-constrained interactions.

An arrow-type semigroupoid is a partially defined associative algebraic structure that encodes the typing, composition, and transformation behavior of arrows (i.e., morphisms) between objects. In these models, arrows are annotated with source and target types (objects), and composition is defined only when types match. Such structures generalize semigroups and categories, capturing the formal underpinnings of typed computations, transformation systems, and certain classes of relational or computational processes. The arrow-type semigroupoid serves both as a canonical reification of type constraints and as a mathematical invariant under various algebraic or categorical constructions, with a variety of roles in automata theory, operator algebras, categorical algebra, and dynamical systems.

1. Formal Definitions and Type-Based Structure

A semigroupoid (or semicategory) is an algebraic structure $(S, s, r, \cdot)$ where $S$ is a set of arrows, $s, r : S \to O$ (for some set $O$ of objects or types) assign a source and range to each arrow, and a partially defined composition $\cdot$ satisfying

$$(a, b) \in S^{(2)} \implies s(ab) = s(b),\;\;\; r(ab) = r(a),\;\;\; (ab)c = a(bc)$$

whenever the compositions are defined. Arrow-type semigroupoids, denoted $S_\rightarrow$, reify the underlying type structure by considering the set of arrow types $(x, y)$, with at most one arrow from $(x,y)$ to $(y,z)$, and the canonical composition: $(x, y) \cdot (y, z) = (x, z).$ This structure captures the combinatorics of type-constrained transformation.

Arrow-type semigroupoids provide a blueprint capturing all placements where composition is possible, regardless of multiplicity of actual arrows between the same types. This is central in the computational exploration of semigroupoids (Egri-Nagy et al., 31 Aug 2025).

2. Homomorphisms: Strict vs. Permissive Variants

Homomorphisms between semigroupoids are structure-preserving maps that encode how type and compositional information is transported between semigroupoids. In the formal setting, a homomorphism $\varphi: S \to T$ must satisfy, for all composable pairs $(a, b)$: $\varphi(a b) = \varphi(a) \varphi(b).$ The theory distinguishes two variants:

• Strict homomorphism: Not only is composition preserved, but so is non-composability—if $(a, b)$ is non-composable in $S$, then $(\varphi(a), \varphi(b))$ is non-composable in $T$. This leads to strong invariance: bijective strict homomorphisms are true isomorphisms of the structure.
• Permissive homomorphism: The requirement is only that the homomorphism is defined on composable pairs; it might collapse or extend compositional possibilities, admitting a weaker form of correspondence. Hence, bijective homomorphisms can exist whose inverses do not preserve the non-composability, deviating from structural isomorphism.

These distinctions have concrete implications. In typed computational systems or semantical models where type consistency enforces security or correctness, strict homomorphisms preserve the integrity of type constraints (Egri-Nagy et al., 31 Aug 2025).

3. Associativity, Typing, and Partiality

Associativity and the existence of a consistent typing (i.e., assignment of source/target to arrows compatible with composition) are logically independent properties. Specifically:

• Associativity: For all $a, b, c$, wherever $(ab)$ and $(bc)$ are defined, must have

$(ab) c = a (bc).$

Non-composable pairs, marked by a null element (e.g., $n$ or \texttt{:n}), must propagate undefinedness according to: - If $ab = n,$ then $a(bc) = n$; if $bc = n,$ then $(ab)c = n$.

• Consistent Typing: Assign $\mathrm{dom}(a)$, $\mathrm{cod}(a)$ to each arrow so that, whenever $ab = c$,

$$\mathrm{cod}(a) = \mathrm{dom}(b),\quad \mathrm{dom}(a) = \mathrm{dom}(c),\quad \mathrm{cod}(b) = \mathrm{cod}(c).$$

Composition is defined only when the types match. These constraints can be encoded as unification goals in logical or declarative programming (Egri-Nagy et al., 31 Aug 2025).

Importantly, there exist partial operation tables that are associative for their domain of definition but do not admit a consistent type decoration (and vice versa). Automated reasoning about semigroupoids must consider both constraints explicitly.

4. Algorithmic Methods for Enumeration and Representation

Enumerating arrow-type semigroupoids and related structures involves several computational strategies:

• Enumeration: Representation as partial composition tables, constructed as vectors with non-composable slots (e.g. \texttt{:n}). Enumeration is achievable via brute-force search or by recursive, logic programming approaches (e.g., using miniKanren or core.logic).
• Associativity/Typing: Implemented as relational goals or constraints, checked for every triple or pair in the table.
• Canonical Form/Isomorphism Check: To minimize redundancies, canonization under vertex permutations is used, discarding isomorphic duplicates.
• Transformational Representation: Embedding abstract semigroupoids into full transformation semigroupoids, by expanding each type to a finite set and assigning arrows to partial functions between these sets. This leverages embedding methods from automata theory (Egri-Nagy et al., 31 Aug 2025).

Such declarative relabeling permits both systematic enumeration of finite arrow-type semigroupoids and construction of representation mappings.

5. Arrow-Type Structures in Broader Algebraic and Categorical Contexts

Arrow-type semigroupoids are not isolated artifacts, but are reflected and generalized in diverse algebraic, categorical, and operator-theoretic frameworks:

• Inverse Semigroup Actions and Groupoids: There is a categorical adjunction between inverse semigroups and étale groupoids, with the inverse semigroup of bisections reflecting the partial symmetries encoded by arrow-type structure (Buss et al., 2011).
• Semigroupoids in Operator Theory: In *-semigroupoids, operator-valued positive semidefinite maps can be dilated to (possibly unbounded) *-representations, connecting to the structure of Cuntz-Krieger-Toeplitz systems, with the arrow system serving as the data of partial isometries (Gheondea et al., 2023).
• Algebra in Arrow Categories: Arrow categories kill off the need for units, focusing on arrays of composable morphisms, and lift algebraic structure (e.g., monoids, bialgebras) in a pointwise manner, paralleling the blueprint of arrow-type semigroupoid composition (Goedicke et al., 2023).
• Object-Free and Relational Semigroupoids: The object-free perspective, where all data is encoded in the arrows and relations, is captured by ternary relations or partial monoids, blurring conventional distinctions between categories and pure semigroupoids (Cranch et al., 2020).
• Quantum and Restriction Semigroupoids: Quantum semigroupoids and Szendrei expansions generalize arrow-type semigroupoids within weak bialgebraic and restriction-theoretic frameworks, providing universal completions and factorizations (Huang et al., 2020, Haag et al., 28 Apr 2025).

6. Applications and Computational Significance

Arrow-type semigroupoids serve multiple roles:

• Canonical Type Invariants: Given a semigroupoid, its arrow-type semigroupoid is an invariant (up to relabeling) of the underlying type structure, capturing the foundational data necessary for composition.
• Models of Typed Processes: The abstract formulation models discrete computational processes, automata, or dynamical flows with explicit typing or resource constraints (Egri-Nagy et al., 31 Aug 2025).
• Classification: Because the enumeration of arrow-type semigroupoids corresponds to enumeration of transitively closed directed graphs (without parallel arrows), their study interacts with graph theoretical problems and the theory of relational data modeling.
• Security and Information Flow: Strict homomorphisms provide insight into the algebraic control of type-preserving computations, important in secure computation, (typed) process calculi, and categorical semantics.

7. Summary Table: Strict vs. Permissive Homomorphisms

Homomorphism Type	Preserves Non-Composability?	Isomorphism = Bijection?
Strict	Yes	Yes
Permissive	No	Not always

A strict homomorphism is structure-reflecting for typed computations; permissive homomorphisms weaken the notion of morphism equivalence.

• Algorithmic, structural, and representation theory: "Computational Exploration of Finite Semigroupoids" (Egri-Nagy et al., 31 Aug 2025)
• Adjunctions and groupoid/symmetry perspectives: "Inverse semigroup actions as groupoid actions" (Buss et al., 2011)
• *-semigroupoids and operator representation: "Partially Positive Semidefinite Maps on $*$-Semigroupoids and Linearisations" (Gheondea et al., 2023)
• Object-free, relational, and categorical algebra: "Relational Semigroups and Object-Free Categories" (Cranch et al., 2020)
• Quantum extensions: "Universal quantum semigroupoids" (Huang et al., 2020)
• Universal completion/representation: "The Szendrei Expansion of Restriction Semigroupoids" (Haag et al., 28 Apr 2025)
• Arrow categories: "On Structures in Arrow Categories" (Goedicke et al., 2023)

Arrow-type semigroupoids thus supply a rigorous and computable foundation for reasoning about the algebraic, categorical, and computational behavior of (partially composable) typed arrows throughout modern mathematics and theoretical computer science.