Lambek Calculus Fragments

• Lambek Calculus Fragments are precisely defined subsystems obtained by restricting logical connectives and structural rules in the full Lambek calculus.
• They employ algebraic semantics, such as pointed semilatticed monoids and ideal completions, to model non-classical, resource-sensitive logics.
• These fragments exhibit strong meta-properties including cut-elimination, decidability under designated weakening rules, and conservative extension of the full system.

Fragments of the Lambek calculus constitute precisely delineated subsystems of the broader Lambek family and its substructural extensions, often isolated by omitting, restricting, or modifying certain logical connectives or structural rules. These fragments play a central role in the paper of proof theory, algebraic semantics, decidability, and expressivity within substructural logics, as well as their applications in formal linguistics and computation.

1. Definition and Classification of Fragments

A fragment, in this context, is any subsystem of the (full) Lambek calculus, typically obtained by restricting the language to a subcollection of connectives (e.g., omitting implications or products), imposing structural constraints (associativity, permutation, etc.), or bounding the syntactic complexity of formulae. The Full Lambek Calculus (FL) itself is a Gentzen system with the following features:

• Syntax: Formulae built from a countable set of variables, binary connectives (multiplicative product $*$, additive conjunction $\wedge$, disjunction $\vee$, divisions $\backslash$, $/$), constants ($0, 1$), and sometimes negations (${}'$, $^\prime$).
• Sequents: $\Gamma \Rightarrow \Delta$ with $\Gamma, \Delta$ finite sequences; $|\Delta|\le 1$.
• Structural rules: Exchange (e), left-weakening ($w_\ell$), right-weakening ($w_r$), contraction (c) may be selectively present or absent.

Implication-free fragments are of particular interest and correspond to subsystems $\operatorname{FL}_\sigma[Y]$ where the set $Y$ excludes implications and possibly other connectives.

Table: Major Classes of Fragments

Fragment Type	Connectives Retained	Key Features
Additive-multiplicative, no impl.	$\vee, \wedge, *, 0, 1$	Algebraizes as semilatticed monoid
Product-free	$\vee, \wedge, 0, 1$	No $*$; meet/disjunction only
Negation fragments	$\vee, *, {}', {}^\prime, 0, 1$	Includes unary pseudocompl.
Combination w/negations	$\vee, \wedge, *, {}', {}^\prime, 0, 1$	Most expressive no-implication fragment

2. Gentzen Systems, Extensions, and Their Fragmentation

The Gentzen system $\operatorname{FL}_\sigma$ is determined by a sequence $\sigma$ of admissible structural rules. Its fragments are defined by restricting the introduction rules to those connectives in a subset $Y$, thus forming $\operatorname{FL}_\sigma[Y]$. Structural rules are:

• (e) Exchange (permutativity on the left)
• ($w_\ell$), ($w_r$) Left/right weakening (integrality, boundedness)
• (c) Contraction (increasing idempotency)

Each fragment admits a corresponding Gentzen system and, when formulated externally (as a Hilbert system), an algebraic semantics.

Key theorems include:

• Fragment Theorem: For any $\sigma$ and fragment $Y$, $\operatorname{FL}_\sigma[Y]$ is the exact $Y$-fragment of $\operatorname{FL}_\sigma$.
• External Correspondence: The external system $\mathcal{E}(\operatorname{FL}_\sigma)[Y]$ is the $Y$-fragment of $\mathcal{E}(\operatorname{FL}_\sigma)$.

3. Algebraic Semantics and Ideal Completion

The algebraic semantics of these fragments is well-developed:

• Pointed semilatticed monoids $(A,\vee,*,0,1)$ form the algebraic models for implication-free fragments with $\vee, *, 0, 1$.
• Residuation (full systems): $a*b\le c \iff b\le a\backslash c \iff a\le c/b$.
• Pseudocomplemented monoids: For negation fragments, the pseudocomplements $a'$ and $'a$ satisfy $a*b\le 0 \iff b\le a' \iff a\le' b$.

Ideal completion is a foundational device: any semilatticed monoid $A$ embeds into the complete algebra of its ideals $\operatorname{Idl}(A)$, with operations extended to sets via closure.

• This embedding is structure-preserving: $\iota_A(a) = (a] = \{x\mid x \le a\}$ preserves $\vee, *, 0, 1$, and residuals where defined.
• Subreduct theorems establish that the semilatticed monoids, pseudocomplemented monoids, and their variants are precisely the reducts (i.e., the images under forgetting operations) of the full variety of (pointed/pseudocomplemented/residuated) lattices.

4. Meta-theoretic Properties: Cut-Elimination, Decidability, Algebraizability

• Cut-Elimination and Subformula Property: Hold in all $\operatorname{FL}_\sigma$ – and their fragments $\operatorname{FL}_\sigma[Y]$ – for any $\sigma$ not containing contraction ($c\notin \sigma$).
• Algebraizability: Each $\operatorname{FL}_\sigma[Y]$ admits an algebraic semantics, but with a crucial caveat:
  • While there exist semantics (e.g., pointed semilatticed monoids), the fragments are not even protoalgebraic and are not equivalent to any Hilbert system or even to the external system $\mathcal{E}(\operatorname{FL}_\sigma)[Y]$.
  • This stands in contrast to the full system, for which external and Gentzen systems are equivalent and algebraizable.
• Decidability: If $w_\ell\in\sigma$ (i.e., some weakening present), then the variety has the Finite Embeddability Property (FEP), leading to decidability of the quasi-equational theory and hence of the consequence relations of the fragment and its algebraic models.
• Fragment/Extension Relations: Every fragment is a conservative sub-system of its parent calculus; external fragments are the restriction of the full external calculus.

5. Algebraic Characterizations and Embeddings

For each major implication-free sublanguage $Y$ and each sequence of structural rules $\sigma$, the corresponding class of algebras (variety) is described by:

• For $Y=\{\vee,*,0,1\}$: pointed semilatticed monoids $\mathcal{Misi}_\sigma$ with structural identities corresponding to $\sigma$.
• For $Y$ including pseudocomplements: pseudocomplemented monoids $\mathcal{PMS}_\sigma$.
• For $Y$ with meet: these expand to semilatticed structures, potentially lattices when both meets and joins are present.

Every such algebra embeds into its ideal completion, a complete (residuated, pointed, or pseudocomplemented) algebra.

Non-protoalgebraicity: Despite admitting an algebraic semantics, the implication-free fragments cannot be axiomatized by equivalence formulas, hence are not (proto)algebraic in the sense of Blok–Pigozzi theory.

6. Decidability, Expressivity, and Logical Landscape

• Additive-multiplicative implication-free fragments behave in a well-understood, decidable fashion—assuming enough weakening.
• With contraction, cut-elimination and subformula property may fail.
• These fragments are strictly less expressive than the full calculus, but enjoy strong meta-properties due to their embeddability into full FL-algebras via ideal completion.
• For each such substructural Gentzen system, the lack of implication impedes full algebraic correspondences in the Hilbert style, reflecting a subtle but consequential break in harmony.
• Varieties classified by $\sigma$ describe algebraic models for specific substructural logics, e.g., commutative, integral, or idempotent semilatticed monoids.
• Soundness and completeness are established w.r.t. the appropriate class of algebras, but only Gentzen systems are strongly complete.

7. Broader Implications and Context

The systematic paper of implication-free fragments reveals key insights into the proof theory and semantics of substructural logics:

• These fragments model non-classical logics where implication is emergent or suppressed, aligning with fuzzy logics (see On et al. 2007) and certain branches of algebraic logic.
• Practical relevance includes modeling computational systems where resource sensitivity (no weakening or exchange) is critical but implication is not basic.
• The lack of protoalgebraicity in Hilbert-style systems signals the necessity of sequent-style reasoning in applications.
• The interplay between algebraic completeness (via ideal completion) and proof-theoretic properties (cut-elimination, subformula property) continues to inform developments in both logic and algebraic semantics.

In conclusion, implication-free fragments of the Full Lambek calculus and its extensions form a robust, algebraically rich, and computationally tractable class of substructural logics, with precisely delineated meta-properties, algebraizations, and model-theoretic behaviors, situated at the intersection of proof theory, algebra, and the algebraic semantics of non-classical logics.