AMPLE AND LEFT AMPLE SEMIGROUPS Extended

When E is actually the set of all idempotents of S, we drop the phrase “with respect to” and refer simply to the left or right ample condition. Recall that a partial permutation of a nonempty set X is a bijection σ : Y → Z for some subsets Y, Z of X, and that the set of all such partial permutations (denoted by IX) is a monoid under the usual composition of partial functions. Note that if σ : Y → Z is a member of IX , then so is its inverse σ −1 : Z → Y so that on IX , we can define three unary operations , † and ∗ as follows: for σ ∈ IX , σ is the inverse of σ; σ = σσ and σ = σσ.