Free Partial Commutative Monoids

• Free partial commutative monoids are algebraic structures built on a set with a partially defined, commutative, and associative binary operation, generalizing classical free monoids under compatibility constraints.
• They are constructed using string rewriting systems that ensure termination and confluence, leading to unique normal forms and well-defined multiplication on compatible pairs.
• These structures find practical applications in trace theory, concurrency models, factorization invariants, and categorical as well as topological constructions in computational research.

A free partial commutative monoid is an algebraic structure generated by a set with a partially defined commutative and associative binary operation, subject to constraints that capture both the “free” generation property and the partiality of multiplication. Such monoids generalize classical free commutative monoids by restricting the multiplication to only defined (compatible) pairs and imposing associativity in a partial fashion. This framework features deep interconnections with string rewriting systems, factorization theory, categorical constructions, and partial semigroup theory.

1. Formal Definition and Structure

A partial monoid $P$ is a nonempty set endowed with a partially defined binary operation $\times$ and a total identity $1_P$. It satisfies:

• Partial associativity: For $x, y, z \in P$, $(x \times y) \times z$ is defined if and only if $x \times (y \times z)$ is defined, and when defined, $(x \times y) \times z = x \times (y \times z)$.
• Identity: For all $x \in P$, $x \times 1_P = x = 1_P \times x$.

The free partial commutative monoid on a set $X$ is constructed by freely generating all finite formal products of elements of $X$, restricting multiplication to compatible pairs, and imposing partial commutativity: $x \times y = y \times x$ whenever both sides are defined. This construction embeds $P$ into the free monoid $P^*$ (i.e., words in $X$), and simulates the partial multiplication as a string rewriting process (Poinsot et al., 2010).

2. Rewriting Systems and Normal Forms

The simulation of the partial operation among generators within the free monoid $P^*$ is achieved by a semi-Thue string rewriting system $R_P$:

• For every $(x, y) \in \mathrm{dom}(\times)$, include the rewriting rule: $i_P(x)\; i_P(y) \to i_P(x \times y)$.
• The identity element is subject to $i_P(1_P) \to \epsilon$, where $\epsilon$ denotes the empty word.

The reduction process proceeds by systematically applying rewriting rules to words in $P^*$, merging adjacent letters when their product is defined, and replacing identity elements with the empty word. The left standard reduction strategy guarantees both termination (word length decreases) and confluence (unique normal forms), which is essential for the well-definedness of multiplication in the quotient monoid of normal forms. The induced multiplication on normal forms, denoted $u \star v = \mathrm{lstd}(uv)$, is strictly associative if and only if $R_P$ is confluent (Poinsot et al., 2010).

3. Partial Commutativity and Factorization Theory

In free partial commutative monoids, the multiplication is commutative only on compatible pairs. This is in contrast to classical free commutative monoids, where all generators commute universally. The partiality of multiplication translates to the rewriting system: only certain letters can merge or commute, depending on the compatibility relation.

Factorization properties are tightly controlled. The monoid is atomic (every non-invertible element decomposes into irreducibles), satisfies the ascending chain condition on principal ideals (ACCP), and often exhibits the greatest common divisor property when suitable conditions are imposed (Jędrzejewicz et al., 2018, Gotti et al., 2022).

The factorization monoid $\mathcal{Z}(P)$ (the free commutative monoid on the set of atoms) organizes the set of all possible atomic decompositions, and factorization invariants such as set of lengths, unions, catenary degree, and elasticity apply mutatis mutandis as in classical commutative monoids, modulo the domain restrictions imposed by partial multiplication (Geroldinger et al., 2019).

4. Connections with Categorical and Topological Constructions

The construction of free partial commutative monoids aligns with categorical presentations using object-free categories and relational semigroup frameworks (Cranch et al., 2020). In this categorical view:

• The partial operation corresponds to a ternary relation that encodes when a product is defined.
• The possibility of multiple units (identities for each object) is reconciled by either imposing extra identifications or by adjoining a zero element to totalize the structure, collapsing units into weak units and making the composition everywhere defined (with 0 as the absorbing element).
• The coherence conditions (e.g., the existence of source and target maps) can be interpreted as requiring commutativity and associativity only on the domain where multiplication is defined.

The concept of Σ-monoids further generalizes addition in commutative monoids to infinite summations; the partial monoid operation $\Sigma: X^* \rightharpoonup X$ is only defined for summable families. The free Σ-monoid admits a left adjoint to the forgetful functor from Hausdorff commutative monoids, and has well-defined tensor products that distinguish it from the classical category of topological abelian groups (Andrés-Martínez et al., 2023).

5. Applications and Examples

Free partial commutative monoids have prominent applications in trace theory, concurrency (Petri nets), linear logic, and resource semantics. In computer science, they model concurrent systems where commutativity is restricted by dependency relations (Poinsot et al., 2010, Gutik et al., 2017). The quotient of certain semigroups of monotone injective partial selfmaps with cofinite domains and images yields free commutative monoids (Gutik et al., 2017).

The theory of square-free elements carries over: every atom (irreducible element) in a free partial commutative monoid is square-free, and decompositions exhibit uniqueness properties as in the factorial case (Jędrzejewicz et al., 2018).

Factorization invariants are tightly controlled; the set of lengths $\mathcal{L}(a)$ for $a \in P$ (encoding the possible numbers of atoms in factorizations) often forms an interval or exhibits almost arithmetical multiprogression structure unless pathological finiteness conditions fail (Cisto et al., 2023, Geroldinger et al., 2019).

6. Generalizations, Self-Similar Group Actions, and Higher-Rank Monoids

The notion of free partial commutative monoids can be generalized via $k$-monoids (higher-rank graphs in $C^*$-algebra theory), where atoms are organized into $k$ types (blocks) and multiplication is prescribed by compatibility rules among blocks (Lawson et al., 27 Mar 2024). Self-similar group actions (Zappa–Szép products) operate on these structures, yielding monoids with unique decompositions, semidirect product presentations, and higher-dimensional analogues of classical Thompson groups.

For a strict $k$-monoid $S$, equipped with a size map $\delta: S \to \mathbb{N}^k$, partial associativity and commutativity persist block-wise; group actions can be encoded geometrically via cubical relations, underpinning both algebraic and topological features of the monoid.

7. Associativity, Confluence, and Algebraic Properties

The equivalence between associativity of the induced multiplication on normal forms and confluence of the rewriting system is the central algebraic assertion (Poinsot et al., 2010):

• If the string rewriting system is confluent (i.e., all reduction paths lead to a unique normal form), then the multiplication is strictly associative.
• Otherwise, associativity holds only modulo the Thue congruence induced by the rewriting rules.

Critical pairs in rewriting systems signal obstructions to confluence and associativity, and their resolution delineates the algebraic structure of the resulting free partial commutative monoid.

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Table: Algebraic Features of Free Partial Commutative Monoids

Feature	Manifestation	Papers
Partial associativity	$(x \times y) \times z$ defined iff $x \times (y \times z)$	(Poinsot et al., 2010)
Partial commutativity	$x \times y = y \times x$ on compatible pairs	(Cranch et al., 2020, Gutik et al., 2017)
Atomicity	Every element factors into irreducibles	(Jędrzejewicz et al., 2018, Gotti et al., 2022)
Factorization invariants	Catenary degree, set of lengths, elasticity	(Geroldinger et al., 2019, Cisto et al., 2023)
String rewriting	Terminating/confluent system defines multiplication	(Poinsot et al., 2010)

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The paper of free partial commutative monoids integrates string rewriting, categorical semantics, factorization theory, and combinatorial presentations, providing foundational tools for algebraic and computational research in partially commutative structures, trace monoids, and generalizations via higher-rank or self-similar constructions.