In an attempt to classify all of the overlap-free morphisms constructively using the Latin-square morphism, we came across an interesting counterexample, the Leech square-free morphism. We generalize the combinatorial properties of the Leech square-free morphism to gain insights on a larger class of both overlap-free morphisms and square-free morphisms.

Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b+ba∗)=Φ(a)∘Φ(b)+Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on B, respectively), for all elements a,b∈A, if only Φ ∗-ring isomorphism. particular, algebras are type I factors, then unitary isomorphism or conjugate

The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme Bn of finite type over Z, with geometrically connected fibers. It is smooth if and only if n ≤ 3. It is reducible if n ≥ 8 (and the converse holds, at least if we remove the fibers above 2 and 3). The relative dimension of Bn is 2 27 n +O(n). The subscheme parameterizing étale algebras is isomorph...

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a “cohomology theory” for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to L-homology. Chern classes of algebraic bundles are defined in the theory. There is a na...