Clifford Semigroup

A semigroup that is both an inverse and a completely regular semigroup . A simple consequence is that a Clifford semigroup is a semilattice of groups. This means that the set of maximal subgroups {S _α: α ∈ A} can be indexed by the members of a commutative semigroup of idempotents A such that and S _α S _β⫅ S _{α β} for each α, β ∈ A. The details of the multiplication in S can be readily deduced. In particular, for each α, β ∈ A with α β = β there exists a homomorphism ϕ_α,β:S _α → S _β. The homomorphisms are such that ϕ_{α, α} is the identity map on S _α, and if α β = β, β γ = γ, then ϕ_{α, β}ϕ_{β, γ} = ϕ_{α, γ}. For a ∈ S _α and b ∈ S _β, the multiplication in S is given by ab = (aϕ_{α, α β})(bϕ_{β, α β}).

For application of Clifford semigroups to the Yang–Baxter equation see weak Hopf algebra .