Schreier-Zassenhaus theorem for algebras. I
Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α −1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.
We introduce a weakened form of regularity, the so called semiregularity, and we show that if every diagonal subalgebra of is semiregular then is congruence modular at 0.