Quasi Relation Algebras (qRAs)

• Quasi relation algebras are algebraic structures defined on distributive lattices with monoid operations and multiple negations that generalize classical relation algebra theory.
• They provide a framework for concrete representations via up-set lattices and duality principles, offering insights into contraction and ordinal sum constructions.
• Their study connects to related structures like directed cylindric algebras, highlighting key open problems in finite representability and structural characterizations.

A quasi relation algebra (qRA) is an algebraic structure that axiomatizes operations on binary relations, extending classical relation algebra theory by relaxing Boolean conditions and introducing multiple negation-like operations. The distributive subvariety, distributive quasi relation algebras (DqRAs), forms a central research topic as they generalize representable relation algebras (RRAs) through distributive lattices and richer involutive structure. DqRAs provide a framework for studying representations, dualities, and structural phenomena in substructural logic and algebra.

1. Algebraic Foundations of Quasi Relation Algebras

A qRA consists of a distributive lattice $\langle A,\wedge,\vee\rangle$, a monoid structure $(A,\cdot,1)$, and a suite of unary operations:

• Residuals $\backslash, /$ defined by the law: $a\cdot b\le c\iff a\le c/b\iff b\le a\backslash c$.
• Left negation: $\sim a := a\backslash 0$
• Right negation: $-a := 0/a$
• Involution (often denoted $'$ or $\neg$): an involutive order-reversing map, i.e., $\neg\neg a = a$ and De Morgan laws $\neg(a\vee b) = \neg a\wedge \neg b$.

A qRA is an involutive FL-algebra satisfying additional interaction and distribution axioms:

• (Di) $\neg(\sim a) = -( \neg a )$
• (Dp) $\neg(a \cdot b) = \neg a + \neg b$, with $a + b := \sim(-b\cdot -a)$

A DqRA restricts the lattice to be distributive, and may further satisfy cyclic conditions ($\sim a = -a$ for all $a$).

2. Representation Theory

Concrete representations of DqRAs use the correspondence with up-set lattices of partially ordered equivalence relations:

• Let $(X,\le)$ be a poset and $E\subseteq X^2$ an equivalence relation with $\le\subseteq E$.
• The up-set lattice $\mathsf{Up}(E)$ under the order $(u,v)\le (x,y)\iff x\le u \land v\le y$ forms a distributive lattice.
• Operations are realized as follows:
  • Monoid identity: $1=\le$
  • Composition: $$R \circ S = \{ (x,y)|\exists z: (x,z)\in R\ \land (z,y)\in S \}$$
  • Residuals: $R\backslash S = (R^\dagger \circ S^c)^c$, $S/R = (R^c \circ S^\dagger)^c$
  • Negations use order automorphisms $\alpha, \beta$: $\sim R = (R^c)^\dagger \circ \alpha$, $-R = \alpha \circ (R^c)^\dagger$, $R' = \alpha \circ \beta \circ R^c \circ \beta$ with symmetry requirements.

Representation theorems establish that DqRAs are representable iff they embed into products of such full up-set algebras for suitable $(X, E, \alpha, \beta)$ (Craig et al., 2023, Craig et al., 9 Mar 2025, Craig et al., 12 May 2025). The representation generalizes that of relation algebras: setting $\alpha=\mathrm{id},\;\beta=\mathrm{id},\;E=X^2$ recovers classical RRA.

3. Structure Theory, Contraction, and Duality

The presence of "positive symmetric idempotents" (PSIs)—elements $p$ with $1\le p$, $p\cdot p=p$, and $\sim p = -p = \neg p$—enables contraction. The contraction $pAp$ is defined by restricting all algebraic operations to $pAp := \{ p\cdot a\cdot p\mid a\in A \}$, taking $p$ as the new identity. Every contraction of a representable DqRA is again representable (Craig et al., 22 Jan 2026).

Duality theory for complete perfect DqRAs involves frames $\mathbb W=(W,I,\preccurlyeq,\circ,\,^{\sim},\,^{-},\,^{\neg})$ with binary and unary operations satisfying analogues of the DqRA axioms. The complex algebra $\mathbb W^+$ constructed from up-sets in such frames yields a DqRA, and every complete perfect DqRA is dual to such a frame (Craig et al., 12 May 2025).

4. Ordinal Sums and Finite Representability

Generalized ordinal sums are a key construction. If $K$ is an odd DqRA (i.e., $0=1$) and $L$ is any DqRA, their sum $K[L]$ remains a DqRA, with operations and negations extended appropriately. Crucially, given representable DqRAs $K,L$, $K[L]$ is again representable under mild additional hypotheses (Craig et al., 9 Mar 2025).

Notably, finite Sugihara chains $S_n$ (distributive, involutive, commutative RLs ordered as chains) are all finitely representable: $S_n\cong S_3[S_{n-2}]$ and each stage preserves representability.

The key finite non-representability criterion, inherited from the classical case, states: If a DqRA has $a$ with $0$a^2\le 0$, then it cannot be finitely represented. Contraction methods generalize this: if some $b$ has $p\cdot b=b=b\cdot p$, $-p<b<p$, and $b^2\le -p$, neither $pAp$ nor $A$ can be finitely represented (Craig et al., 22 Jan 2026, Craig et al., 12 May 2025).

5. Examples, Classification, and Known Catalogues

Enumerative work using Mace4/Prover9 and dual-frame counting yields a comprehensive catalogue for DqRAs of small size (Craig et al., 12 May 2025): | Size | Number of non-isomorphic DqRAs | Known Representability | |------|-------------------------------|------------------------| | 1 | 1 | Yes | | 2 | 1 | Yes | | 3 | 2 | ${D^3_{1,2}}$ (Sugihara) representable; ${D^3_{1,1}}$ not known finitely representable | | 4 | 10 | Only one non-cyclic representable both finitely and infinitely; remainder open or non-representable | | 5 | 8 | Various; see explicit tables in (Craig et al., 12 May 2025) | | 6 | 50 | Several explicit non-representable examples identified via contraction and "small square" criteria |

Key examples:

• The four-element Sugihara chain is representable on a two-element poset with identity automorphisms; $\sim$ and $-$ coincide and $\neg$ behaves classically.
• The three-element chain $D^3_{1,1}$ with $0$a^2=0$ is non-representable by the finite square criterion—no finite poset yields a valid up-set representation.
• Chains and diamonds yielding non-cyclic DqRAs often fail finite representability except in specific cases, as confirmed by explicit computation.

6. Connections to Other Algebraic Structures

Quasi-projective relation algebras (QPRAs) and distributive quasi relation algebras are connected via categorical equivalence to directed cylindric algebras of dimension 3 (DCA$_3$) (Ahmed, 2013). Functors $F:\mathrm{QRA}\to\mathrm{DCA}_3$ and $G:\mathrm{DCA}_3\to\mathrm{QRA}$ are order-preserving and inverse up to isomorphism. The superamalgamation property is inherited between these categories, establishing strong structural parallels.

Gödel’s second incompleteness theorem is valid within the QRA framework, with finite-variable encodings, pairing techniques, and algebraic reflection of consistency statements as developed in (Ahmed, 2013).

7. Open Problems and Research Directions

Research remains active in several core questions:

• Whether every DqRA is representable (i.e., does $\mathrm{RDqRA} = \mathrm{DqRA}$?).
• Whether the class $\mathrm{RDqRA}$ is a variety, as is known for RRAs.
• The possibility of transferring Maddux’s sufficient conditions for RRA representability to the quasi setting.
• Establishing game-based hierarchies for representability as in classical relation algebra theory.
• Developing concrete representation and duality results for non-distributive qRAs.

The interplay between contractions, duality theory, ordinal sums, and the catalogue of small DqRAs provides a substantial foundation for future investigation, particularly on the boundaries of finite and infinite representability and the expressivity of negation operations.

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Citations: (Craig et al., 2023, Craig et al., 9 Mar 2025, Craig et al., 12 May 2025, Craig et al., 22 Jan 2026, Ahmed, 2013)