A Single-Point Reduction and Squaring Endomorphism Reformulation of E677 ⊨_fin E255
Year: 2025
Type: preprint
Abstract: We study finite magmas satisfying the identity E677: a = b ◇ (a ◇ ((b ◇ a) ◇ b)) and the open question of whether every such magma also satisfies E255: ((x ◇ x) ◇ x) ◇ x = x. Using the left-orbit structure and a recurrence relation on the sequence d_k = c_k ◇ x (where c_k = L_x^k(x)), we prove that E255 is equivalent to a single self-squaring identity: c_{p−4} ◇ c_{p−4} = c_{p−5}, where p is the left-orbit period of x. This reduction unifies previously observed mechanisms (universal self-squaring at period 6, Gauss pairing at period 7) into one target identity, verified across 1200+ orbits at size 7 and 30 orbits at size 35. We further show that idempotent elements satisfy E255 trivially, and that the 496-element blueprint construction is entirely idempotent and therefore irrelevant to the non-trivial case.
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160410324