Higher Convolution Kleene Algebras

• Higher convolution Kleene algebras are algebraic structures that lift compositional objects to powerset or function algebras, enabling multi-dimensional convolution products.
• They translate relational and categorical compositions into algebraic operations via convolution, using lax interchange to manage higher-dimensional interactions.
• The theory integrates Möbius conditions, graded decompositions, and modal operators to address iteration challenges and reconcile sequential with parallel composition.

Higher convolution Kleene algebras are convolution-based Kleene-algebraic structures in which a compositional object $C$—such as a relational monoid, catoid, strict higher category, higher path category, or polygraphic cell complex—is lifted to a powerset or function algebra $K^C$ carrying one or more products, modal operators, and iteration operations. The expression is used explicitly for convolution $n$-Kleene algebras on strict higher categories and higher relational monoids in "Generalised Möbius Categories and Convolution Kleene Algebras" (Cranch et al., 29 Aug 2025). Closely related constructions were developed earlier under the names interchange or concurrent Kleene algebra (Cranch et al., 2020), higher globular Kleene algebra (Calk et al., 2020), and convolution $\omega$-quantales with semiring and Kleene-algebra specialisations (Calk et al., 2023). The common principle is that decomposition laws on $C$ induce algebraic composition on predicates, sets, or weighted functions, while higher-dimensional interaction is controlled by lax interchange rather than strict equality.

1. Convolution as the basic lifting principle

The starting point is ordinary convolution on a function space. If $C$ carries a ternary relation $R^x_{yz}$ or, equivalently, a partial or set-valued composition $x\in y\odot z$, and if the value algebra $Q$ has a multiplication, then convolution is defined by summing or joining over all decompositions of $x$. In quantalic form,

$K^C$0

whereas in semiring or Kleene-algebra settings one uses finite sums,

$K^C$1

This is the general mechanism by which object-level composition on $K^C$2 becomes algebraic multiplication on $K^C$3 or $K^C$4 (Cranch et al., 2020).

The Boolean case $K^C$5 yields the powerset or complex-algebra lifting: $K^C$6. In that case convolution is existential composition of subsets. Weighted semantics arise when $K^C$7 or $K^C$8 is a quantale, dioid, semiring, or Kleene algebra, so the same decomposition structure on $K^C$9 produces weighted languages, weighted graph languages, weighted pomset languages, or weighted higher-cell semantics (Cranch et al., 2020).

The higher version replaces one composition by many. In a $n$0-dimensional or interchange setting, $n$1 carries two compositions or ternary relations, typically interpreted as sequential and parallel. In an $n$2-dimensional or $n$3-dimensional setting, $n$4 carries a family $n$5 or $n$6, and $n$7 inherits one convolution product $n$8 per dimension. This is the sense in which higher convolution is not a single operation but a dimension-indexed family of convolution products (Calk et al., 2023).

2. Algebraic signatures: interchange, concurrency, and higher dimensions

The earliest systematic precursor is the theory of interchange and concurrent Kleene algebras. In the two-product setting, one works with a sequential product and a parallel product satisfying the lax interchange law

$n$9

or, in dimension-indexed notation,

$\omega$0

An interchange Kleene algebra consists of two Kleene algebra structures on the same carrier satisfying this law; a concurrent Kleene algebra is the commutative-parallel special case (Cranch et al., 2020).

This two-dimensional pattern was generalized to higher-dimensional algebra in "Algebraic coherent confluence and higher globular Kleene algebras" (Calk et al., 2020). There an $\omega$1-dioid is a family

$\omega$2

such that each $\omega$3 is a dioid and, for $\omega$4,

$\omega$5

An $\omega$6-Kleene algebra adds a star $\omega$7 in each dimension, while globular modal structure adds domain and codomain maps $\omega$8 satisfying globularity laws such as

$\omega$9

for $C$0 (Calk et al., 2020).

A parallel foundational line is developed with $C$1-catoids and $C$2-quantales. An $C$3-catoid is a family $C$4 of catoid structures satisfying source-target compatibility, globular laws, and higher interchange inclusions

$C$5

An $C$6-quantale is the corresponding complete lattice structure with dimension-indexed multiplications, units, domain and codomain operators, and the analogous algebraic inequalities. These constructions were introduced to replace earlier ad hoc higher Kleene-algebra axioms by representation-theoretic ones (Calk et al., 2023).

The terminology is not uniform across the literature. The $C$7 concurrency paper emphasizes interchange and concurrent Kleene algebras (Cranch et al., 2020), the coherent-confluence paper emphasizes higher globular modal $C$8-Kleene algebras (Calk et al., 2020), and the $C$9 Möbius-catoid paper uses the phrase higher convolution Kleene algebras explicitly for the function-space constructions on strict higher categories and higher relational monoids (Cranch et al., 29 Aug 2025).

3. Structural carriers: relational monoids, catoids, higher categories, and Möbius conditions

The carrier of a higher convolution Kleene algebra is not itself a Kleene algebra; it is a compositional structure whose factorisations drive convolution. In the concurrency setting, the relevant objects are relational monoids and relational interchange monoids. A relational interchange monoid $C$0 is a set with two ternary relations, each relationally associative and unital, together with a relational interchange law corresponding to algebraic interchange (Cranch et al., 2020).

The catoid formalism packages the same idea in single-sorted algebraic form. A catoid $C$1 has a set-valued composition and source/target maps satisfying multirelational associativity, weak locality, and unit laws. Categories are local functional catoids. Higher catoids, including $C$2-catoids, $C$3-catoids, and $C$4-catoids, carry one such structure in each dimension, linked by higher source-target and interchange laws. Strict $C$5-categories are the local functional special case, while higher relational monoids are the non-functional, non-local generalization (Calk et al., 2023).

For semiring and Kleene-algebra convolution, the decisive obstacle is iteration. The $C$6 theory resolves this by introducing Möbius catoids. A Möbius catoid combines finite $C$7-decomposability with a finite-length condition: every element has only finitely many relevant decompositions, and recursive definitions can descend along a length function $C$8. Proposition 3.4 of that work characterizes Möbius catoids by finite $C$9-decomposability, indecomposability of identities, and the condition $R^x_{yz}$0 (Cranch et al., 29 Aug 2025). The same paper extends the notion to Möbius $R^x_{yz}$1-catoids and local Möbius $R^x_{yz}$2-catoids of finite valency, which are precisely the higher carriers needed for convolution $R^x_{yz}$3-Kleene algebras.

A further extension is the $R^x_{yz}$4-setting, where dimensions above $R^x_{yz}$5 are groupoidal. An $R^x_{yz}$6-catoid equips higher-dimensional cells with inverses above some dimension, while the algebraic side uses Dedekind quantales or converse-equipped dioids to model homotopic reasoning in higher rewriting (Calk et al., 2023). This broadens higher convolution from purely compositional structure to settings with invertibility and proof equivalence.

4. Star and iteration

In quantales, iteration is straightforward: for each multiplication one defines

$R^x_{yz}$7

This yields stars automatically in convolution quantales, including dimension-indexed stars in $R^x_{yz}$8-quantales and related higher structures (Calk et al., 2023). The same idea appears in interchange quantales, where each product induces its own Kleene star by countable join of powers (Cranch et al., 2020).

The more difficult problem is a genuine Kleene star on $R^x_{yz}$9 when only finite sums are available. In the graded relational setting, an early solution was given for a single product on a graded, finitely decomposable relational monoid with unit $x\in y\odot z$0: $x\in y\odot z$1 This yields a Kleene algebra structure on $x\in y\odot z$2, and then the two-product interchange theorem lifts it to interchange Kleene algebras (Cranch et al., 2020).

The $x\in y\odot z$3 Möbius-catoid construction generalizes the classical Kuich–Salomaa star for formal power series from free monoids to categories with many objects, relational monoids, strict higher categories, and higher relational monoids. For a Möbius catoid $x\in y\odot z$4 and Kleene algebra $x\in y\odot z$5, the star on $x\in y\odot z$6 is defined recursively by

$x\in y\odot z$7

$x\in y\odot z$8

Theorem 5.2 states that if $x\in y\odot z$9 is a Möbius catoid and $Q$0 a Kleene algebra, then $Q$1 is a convolution Kleene algebra with this star (Cranch et al., 29 Aug 2025). Corollary 7.2 gives the $Q$2-dimensional interchange case, and Theorem 8.4 gives the higher $Q$3-dimensional case for local Möbius $Q$4-catoids of finite valency.

Higher globular Kleene algebra approaches iteration differently. There the carrier is usually a powerset algebra over higher cells, and each dimension has its own star $Q$5; extra whiskering and globularity laws control the interaction between stars across dimensions, for example

$Q$6

This is the higher-dimensional analogue of compatibility between iteration and contextual composition (Calk et al., 2020).

An adjacent proof-theoretic line separates two regimes for iteration in noncommutative residuated settings: a fully infinitary, $Q$7-continuous regime and a cyclic regime corresponding to general residuated Kleene algebras not assumed $Q$8-continuous (Kuznetsov, 2017). This suggests a corresponding distinction for higher convolution settings whenever residuals are incorporated, although that extension is not itself developed there.

5. Canonical models and semantic domains

Weighted words and shuffle languages are the standard $Q$9-dimensional example. On $x$0, sequential composition is concatenation and parallel composition is shuffle. Convolution then yields

$x$1

with unit $x$2. If the parallel product in the value algebra is commutative, the resulting weighted shuffle languages form a concurrent Kleene algebra (Cranch et al., 2020).

Structured concurrent objects are treated in the same way. Serial and parallel composition of digraphs induce weighted graph languages; the graph construction specializes to partial orders; and passing to isomorphism classes of finite digraphs yields total composition on graph types. Pomsets arise as isomorphism classes of labelled finite partial orders, so Boolean or weighted pomset languages become canonical models of concurrent convolution Kleene algebra (Cranch et al., 2020). The $x$3 paper explicitly includes free monoid plus shuffle catoids, finite directed graphs, finite posets, pomsets, and higher relational monoids among its examples (Cranch et al., 29 Aug 2025).

A distinct but closely related semantic line comes from higher globular rewriting. For a polygraph $x$4 and cellular extension $x$5, the carrier

$x$6

is the powerset of higher cells, and setwise composition is lifted from higher categorical composition: $x$7 Proposition $x$8 states that $x$9 is an $K^C$00-Boolean $K^C$01-modal Kleene algebra (Calk et al., 2020). The paper does not call this convolution, but structurally it is a complex-product or powerset-lifting construction of exactly the same kind.

Higher-dimensional automata provide another semantic template. The Kleene theorem for HDAs characterizes regular languages as rational subsumption-closed sets of finite interval ipomsets with interfaces, with language operations $K^C$02, gluing composition $K^C$03, parallel $K^C$04, and plus $K^C$05 (Fahrenberg et al., 2022). Gluing is defined by identifying matching source and target interfaces, so it behaves like composition along a common boundary rather than ordinary concatenation. The weak interchange law

$K^C$06

is explicitly used in the tensor-product proof (Fahrenberg et al., 2022). A plausible implication is that interface-sensitive higher convolution products can be read as boundary-based analogues of ordinary language convolution.

6. Correspondence theory, modal structure, and open problems

A defining feature of higher convolution Kleene algebra is that convolution is not merely a construction but a correspondence mechanism. The concurrency paper proves that relational laws on $K^C$07 and algebraic laws on $K^C$08 correspond in the sense of modal logic and Boolean algebras with operators. If $K^C$09 is a relational interchange monoid and $K^C$10 an interchange quantale, then $K^C$11 is an interchange quantale; conversely, under mild nondegeneracy assumptions, algebraic interchange in $K^C$12 recovers interchange laws on $K^C$13 or $K^C$14 via delta functions (Cranch et al., 2020).

The higher catoid/quantale program extends this to full higher-dimensional correspondence triangles

$K^C$15

If $K^C$16 is a local $K^C$17-catoid and $K^C$18 an $K^C$19-quantale, then $K^C$20 is an $K^C$21-quantale; conversely, sufficiently supported convolution algebras reconstruct the higher catoid or the base quantale (Calk et al., 2023). In the modal setting, domain and codomain operators lift by

$K^C$22

These correspondences are explicitly related to Jónsson–Tarski-style dualities between relational structures and lattices with operators (Calk et al., 2023).

The interaction between higher composition and modal structure is central in applications to rewriting and verification. In globular modal $K^C$23-Kleene algebra, coherent Church–Rosser and Newman lemmas are proved entirely by equational reasoning, using stars in multiple dimensions, whiskering, globularity, and weak interchange (Calk et al., 2020). In the $K^C$24 Möbius-catoid framework, modal convolution Kleene algebras, convolution Kleene algebras with tests, concurrent convolution Kleene algebras, and higher convolution Kleene algebras all arise from the same recursive star construction (Cranch et al., 29 Aug 2025).

Several limitations are explicit. Recursive star does not apply to all catoids; pair groupoids, and therefore weighted relations or matrices in general, lack the required length structure (Cranch et al., 29 Aug 2025). Finite decomposability, grading, Möbiusness, or finite valency are often necessary, and in higher-dimensional rewriting finite valency may exclude cyclic behavior and can be restrictive (Cranch et al., 29 Aug 2025). The $K^C$25-catoid paper states that a satisfactory general convolution-star construction for weighted $K^C$26-Kleene algebras with multiple units is not fully solved beyond powerset and quantalic settings (Calk et al., 2023). The concurrency paper explicitly identifies extension of Stone-type duality from the Boolean atomic case to non-atomic quantales and arbitrary convolution algebras, formalisation of the concurrency extension in proof assistants, and a categorification of the approach as future directions (Cranch et al., 2020).

A recurrent misconception is that higher convolution should satisfy strict interchange because the underlying higher categories often do. The algebraic frameworks consistently use lax interchange inequalities instead. This avoids Eckmann–Hilton collapse and preserves the distinction between sequential and parallel, or lower- and higher-dimensional, composition (Cranch et al., 2020). Another misconception is that powerset models exhaust the subject. The later literature shows that weighted, modal, concurrent, and higher-dimensional function-space constructions require additional finiteness and recursion principles, not merely Boolean lifting (Cranch et al., 29 Aug 2025).