A Short Proof of the Irreflexivity Conjecture

and proved that < is a linear ordering. Laver’s proof uses a large cardinal assumption that is not provable in ordinary mathematics. In a series of papers [ ], Dehornoy showed (without any additional assumption) that any distinct a, b ∈ A are comparable in < and reduced the word problem to the problem of existence of a distributive one-generated algebra A in which a < a does not hold for any a ∈ A, the irreflexivity conjecture. A consequence of irreflexivity is that the algebra is free. Dehornoy’s latest paper [ ] proves the irreflexivity conjecture. A major step of Dehornoy’s proof is is the idea of embedding the free distributive algebra into the braid group B∞ (see (5) below). In this note we give a short proof of Dehornoy’s theorem by giving a simple argument showing that the operation ∗ in B∞ satisfies irreflexivity of < . Let F be the free group on free generators x1, x2, x3, . . . , and let B be the group of automorphisms of F generated by {σ1, σ2, σ3, . . .}, where