Extension of GE‐Algebras to NeutroGE‐Algebras

GE-algebras, introduced by Bandaru, Borumand Saeid and Jun as a generalization of Hilbert algebras, are implication-type algebras (X, ∗, 1) defined by three short axioms. This paper presents them in order of increasing difficulty and extends them to neutrosophic and NeutroAlgebraic settings. We start from the smallest examples (the two-element Boolean implication, the three-element chain, the Gödel implication on [0, 1] and a five-element algebra in which the induced relation is not antisymmetric), show by an explicit counterexample that the Łukasiewicz implication on [0, 1] is not a GE-algebra, and prove that a residuated implication gives a GE-algebra exactly when its conjunction is idempotent. We then prove a transport theorem and use it to build neutrosophic GE-algebras on triples (T, I, F), refined neutrosophic GE-algebras on (T₁, …, Tₚ; I₁, …, Iᵣ; N₁, …, N_q; F₁, …, Fₛ), and coupled versions in which the indeterminacy and falsity of an implication depend on the truth values; their orders are bounded distributive lattices. Finally, in the setting of Smarandache's NeutroAlgebra, we show that GE3 is never an AntiAxiom, so every groupoid satisfying GE1 and GE2 is either a GE-algebra or a NeutroGE-algebra; we bound the share of tuples satisfying each subclass identity (sharply for commutativity and antisymmetry), and we classify the instances of GE3 in partial Cayley tables as necessarily true, indeterminate or necessarily false, which yields a neutrosophic triple (T, I, F) for every partial table.