Basis-Cardinality Invariance in Fuzzy Closure Systems: Computational Evidence and Algebraic Characterisation

Abstract Basis-cardinality invariance is a structural property of closure systems whose dependence on the underlying algebraic order remains insufficiently characterised in the fuzzy setting. This paper examines the phenomenon in finite fuzzy closure systems over integral commutative residuated lattices through exhaustive computational enumeration and algebraic analysis. Three complementary investigations are reported. An exhaustive enumeration across nine finite algebra–universe configurations identifies 61 closure systems with unequal-sized inclusion-minimal bases among 173 systems examined. A separate investigation of six finite Heyting algebras identifies 23 such systems among 72. More significantly, a singleton-generator analysis across 15 finite algebras produces 30 witnesses in all 12 tested non-chain cases, while none of the three chain controls exhibits this behaviour. Independent verification of 346 explicit bases yields no discrepancies. Beyond enumeration, a sufficient algebraic criterion is established for the existence of singleton-generated closure systems admitting minimal bases of cardinalities one and two. A further algebraic identity, satisfied by all 30 experimentally identified singleton-generator witnesses, provides a common representation of their generation mechanism. The results establish a consistent computational distinction between the tested chain and non-chain structures and identify an algebraic mechanism underlying the observed failure of basis-cardinality invariance. They do not establish the universal converse for finite non-chain residuated lattices. Instead, they provide a verified computational foundation and a precise structural framework for its investigation.