Isomorphism Theorems for Semigroups of Order-preserving Partial Transformations

The full order-preserving transformation semigroup, the orderpreserving partial transformation semigroup and the order-preserving one-to-one partial transformation semigroup on a poset X are denoted by OT (X), OP (X) and OI(X), respectively. It is well-known that for any posets X and Y , OT (X) ∼= OT (Y ) if and only if X and Y are either order-isomorphic or order-anti-isomorphic. The purpose of this paper is to show the analogous results for OP (X) and OI(X). It is shown that for any posets X and Y , OP (X) ∼= OP (Y ) [OI(X) ∼= OI(Y )] if and only if X and Y are either order-isomorphic or order-anti-isomorphic. From our given proofs, some isomorphism theorems on other order-preserving transformation semigroups are also provided. Mathematics Subject Classification: 20M20