An Optimal 14-Symbol Hybrid Basis for BCH-Algebras

Abstract: We present an optimally minimal two-axiom basis for BCH-algebras. The standard presentation of a BCH-algebra relies on three axioms: two equations and one quasi-identity. Using automated theorem proving, we prove that the two standard equations can be entirely replaced by a 14-symbol equation, ((xy)z)((x(z0))y) = 0, while retaining the standard quasi-identity. We then provide a rigorous proof of strict minimality for this new equational companion. By employing an exhaustive, machine-assisted search space generation coupled with finite countermodel building, we demonstrate that no equation of 12 or fewer symbols can define the class of BCH-algebras when paired with the standard quasi-identity. Our literature searches have revealed no prior proof of this result, to the extent of our knowledge. All equivalence derivations were verified using Prover9, and all minimality countermodels were generated using Mace4.

• The paper introduces a minimal 14-symbol axiom that, with a quasi-identity, replaces the traditional BCH-algebra equations.
• It employs automated tools like Prover9 and Mace4 to rigorously establish the axiom's optimality through exhaustive countermodel searches.
• The results highlight fundamental limits in axiomatic compression for non-varietal algebraic structures, advancing comparative algebraic logic research.

An Optimal 14-Symbol Hybrid Basis for BCH-Algebras

Motivation and Context

The paper addresses axiom compression within the class of BCH-algebras, generalizing prior work on minimal axiom sets for algebraic structures. Given that BCK- and BCI-algebras can be presented with short bases but are not varieties, they require at least one quasi-identity as part of any axiom system. The focus is on optimally reducing the equational component, given the ineliminability of the quasi-identity, thus seeking an optimal “hybrid basis” as opposed to a “pure single axiom” formulation.

The research builds upon the theoretical and technical foundation established by McCune’s single-axiom results for Boolean algebras and leverages state-of-the-art automated reasoning, integrating equational proof search (Prover9) and model search (Mace4) for both positive and negative results.

Main Results and Contributions

The primary outcome is the replacement of the conventional pair of equations for BCH-algebras—$xx = 0$ and $(xy)z = (xz)y$—with a single 14-symbol equation:

$((xy)z)((x(z0))y) = 0$

coupled with the standard quasi-identity:

$xy = 0 \,\wedge\, yx = 0 \implies x = y$

This minimal basis is proven axiomatically equivalent to the standard three-axiom system. Both directions are formally reconstructed:

• The new equation is derivable from the old axioms.
• The old equations $(xx=0, (xy)z=(xz)y)$ are derivable from the new 14-symbol equation plus the quasi-identity.

A rigorous, exhaustive, machine-verified search demonstrates that no equation of fewer than 14 symbols can replace the standard equations when paired with the quasi-identity. All candidate equations of 12 or fewer symbols were eliminated by explicit finite countermodel construction, with no unverified cases or ambiguity.

This establishes that the new 14-symbol axiom is strictly minimal: any shorter equational axiom is either inconsistent or insufficiently expressive for BCH-algebras in combination with the quasi-identity.

Methodology and Technical Approach

Proof Strategy

Both directions of equivalence are addressed structurally:

• The derivability of the new axiom from the old basis proceeds by a sequence of lemmas, leveraging right-commutativity and the behavior of the constant $0$, ultimately reducing to standard identities and exploiting left-associativity.
• Recovery of $xx=0$ and $(xy)z=(xz)y$ from the new system uses paramodulation and symmetry breaking, with critical reliance on quasi-identity reasoning to identify elements uniquely.

The essential role of the quasi-identity is highlighted by recalling that BCH-algebras, like BCK-algebras, do not admit purely equational axiomatization, as proven in [6].

Automated Search and Countermodel Construction

A comprehensive automated search was executed as follows:

• All candidate equations with up to 7 variable or constant leaves (maximal length 12 symbols), with all possible binary tree topologies, were generated.
• Each candidate was tested for soundness (admitted as a theorem in BCH-algebras; otherwise, a finite countermodel was exhibited).
• Remaining candidates were tested for completeness (paired with the quasi-identity, could they replace the standard semantic expressiveness, again witnessed or refuted by finite countermodels).

The absence of “short” equations passing both tests confirms the sharp minimality of the 14-symbol boundary established by the new axiom.

Implications and Future Directions

The results have several implications:

• Axiomatic compression: The strict lower bound on equation length suggests a fundamental limitation in compactly presenting the algebraic structure of BCH-algebras and points to intrinsic complexity arising from their interactions between equations and quasi-identities.
• Automated reasoning: The methodological pipeline—systematic topology enumeration, symmetry pruning, and exhaustive model checking—sets a high standard for mechanized investigations into axiom minimality in non-varietal algebraic systems.
• Comparative algebraic logic: The result situates BCH-algebras alongside other well-studied logical algebraic structures (Boolean, BCK, BCI, etc.) with respect to the compressibility of their defining equations, and invites speculation about similar lower bounds for larger or more complex systems.
• Resource for the community: The open sharing of all verification scripts and search code on Zenodo ensures strict reproducibility and provides infrastructure for further research into axiom compression, potentially in a broader context (e.g., other non-varieties).

Future research might target:

• Extending axiom compression techniques to other varieties and non-varieties in algebraic logic.
• Characterizing the algorithmic complexity of minimal basis discovery as structure size or arity increases.
• Investigating whether the methods scale to yet richer algebraic structures (e.g., multi-sorted or higher-type algebras).

Conclusion

The paper establishes a uniquely minimal hybrid axiom system for BCH-algebras, comprising a single 14-symbol equation and one irreducible quasi-identity, proven by both formal and computational means to be strongly minimal among all possible bases of this form. This work both clarifies the algebraic structure of BCH-algebras and exemplifies the state of the art in automated equational logic, with broader implications for automated axiom discovery and the limits of algebraic compression for logic-algebraic systems.