Universal invariants and value-reflection symmetry in the lattice of R1​-precomplete subclasses of (k+1)-valued logic

Post's lattice fully describes the closed classes of Boolean functions, but its (k+1)-valued analogue is, in general, intractable: the lattice of clones in P_k has the cardinality of the continuum for every k >= 2. Within the well-defined sublattice of R_1-precomplete subclasses inside the class T_01 of functions preserving both 0 and k, we identify a structural invariant which is uniform in k and which we interpret as a universal property of the R_1-closure.

We work within the Pol-Inv Galois, restricted to the R_1-closure operator. The chain {0, ..., k} carries a natural value-reflection involution x -> k - x; its induced action on the subclasses of T_01 is analysed orbit by orbit. The R_1-closure-equivalence classes are enumerated via a matrix-logic representation.

We obtain three results. (i) For every k >= 2 and every n >= 1, the number of R_1-closure equivalence classes of T^(n) equals 2^(2^n - 1), independently of k. (ii) The cells of the structural decomposition satisfy |A_J| = (k+1)^(n - |J|). (iii) A matrix-logic reduction T_sim = P_2 holds, yielding a two-valued shadow of the (k+1)-valued structure. All three statements were cross-verified by exhaustive enumeration for k = 2, 3.

The k-uniformity of the count 2^(2^n - 1), combined with the value-reflection involution, provides a symmetry-type invariant of the lattice. Open directions include the complete orbit poset under the value-reflection involution and a possible polyhedral, in particular tropical, interpretation of the R_1-closed sublattice.