Regularity and isomorhism theorems of some order-preserving transformation semigroups

Abstract

For a poset X, let OT(X), OP(X) and OI(X) denote respectively the full order-preserving transformation semigroup on X, the order-preserving partial transformation semigroup on X and the order-preserving one-to-one partial transformation semigroup on X. The following facts of regularity of order-preserving transformation semigroups are known. For any subchain X of Z, OT(X) is regular, and for an interval X in IR , OT(X) is regular if and only if X is closed and bounded. The semigroups OP(X) and OI(X) are regular for any chain X. An interesting isomorphism theorem of full order-preserving transformation semigroups is that for posets X and Y, OT(X) [is equivalent to] OT(Y) if and only if X and Y are either order-isomorphic or anti-order-isomorphic. Our purpose is to give more results of regularity and isomorphism theorems of order-preserving transformation semigroups. First, we show that for a nontrivial interval X in a subfield F of IR , OT(X) is regular if and only if F = IR and X is closed and bounded. Next, the following respective subsemigroups of OT(X), OP(X) and OI(X) are considered. OT(X, X') = {[alpha][is an element of] OT(X) ran [alpha] [is less than or equal to] X'}, OP(X, X') = {[alpha] [is less than or equal to] OP(X) ran [alpha] [is less than or equal to] X'} and OI(X, X') = {[alpha] [is an element of] OI(X) ran [alpha] [is less than or equal to] X'} where X' is a subchain of a chain X. We characterize when OT(X, X') is regular in terms of X, X' and the regularity of OT(X). It is proved that X = X' is necessary and sufficient for OP(X, X') and OI(X, X') to be regular. The interesting isomorphism theorems of order-preserving transformation semigroups obtained in this research are as follows: If OT(X, X')[is equivalent to] OT(Y, Y'), then X' and Y' are either order-isomorphic or anti-order-isomorphic. If OP(X, X') [is equivalent to] OP(Y, Y'), then X' = Y' and X' and Y' are either order-isomorphic or anti-order-isomorphic. Moreover, forX' > 1 and ' > 1, OI(X, X') [is equivalent to] OI(Y, Y') if and only if there is an order-isomorphism or an anti-order-isomorphism [theta] : X -> Y such that X'[theta] = Y' . Our first isomorphism theorem is an extension of the above known isomorphism theorem for the case of chains. We also show that the converses of our first two isomorphism theorems are not generally true. However, interesting consequences of these two isomorphism theorems are as follows: For any chains X and Y, OP(X )[is equivalent to] OP(Y ) [OI(X ) [is equivalent to] OI(Y )] if and only if X and Y are either order-isomorphic or anti-order-isomorphic.