Regular Elements of Some Semigroups of Order-Preserving Partial Transformations
Winita Mora, Yupaporn Kemprasit · 2010
Let X be a chain, OP(X) the order-preserving partial transformation semigroup on X and OI(X) the order-preserving 1–1 partial transformation semigroup on X. It is known that both OP(X) and OI(X) are regular semigroups. We extend these results by characterizing the regular elements of the semigroups OP(X, Y),OI(X, Y), OP(X, Y) and OI(X, Y) where ∅ = Y ⊆ X, OP(X, Y) = {α ∈ OP(X) | ran α ⊆ Y}, OP(X, Y) = {α ∈ OP(X) | (dom α ∩ Y)α ⊆ Y}, OI(X, Y) and OI(X, Y) are defined analogously. The semigroups OP(X, Y) and OP(X, Y)[OI(X, Y), OI(X, Y)] may be counted as generalizations of OP(X)[OI(X)]. In addition, it is shown that each of these semigroups becomes a regular semigroup only the case that Y = X.