A Note on the Algebraic and Combinatorial Properties of {-1,0,+1}^3

Abstract

This note documents basic properties of the set L_3 = {-1,0,+1}^3, a finite set of 27 integer lattice points. We describe: (1) two binary operations: saturated addition ⊕ (non-associative) and balanced product ⊗ (associative, commutative), with ⊗ distributing over ⊕; (2) the graph automorphism group Aut(L_3) ≅ S_3 ⋉ (Z_2)^3 of order 48, with orbit decomposition 1+6+12+8; (3) 49 collinear triples (magic-sum lines) whose 49×27 incidence matrix over GF(3) has rank 23 and a 4-dimensional null space spanned by {1, x, y, z}; (4) the adjacency graph has 27 vertices, 54 edges, first Betti number β_1 = 28, and Euler characteristic χ = -27; (5) componentwise order yields a distributive lattice with rank distribution 1-3-6-7-6-3-1. All results are elementary and obtained by direct computation or standard corollaries. No claim of original mathematical discovery is made; this note serves as a reference compilation.

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