Order-preserving generalized transformation semigroups

Abstract

For a set X, let P(X), T(X) and I(X) denote respectively the partial transformation semigroup on X, the full transformation semigroup on X and the 1-1 partial transformation semigroup on X. These transformation semigroups are generalized as follows: For sets X and Y, let P(X, Y) = {[alpha]:A-> Y [is less than or equal to] X}, T(X, Y) = {[alpha] [is an element of] P(X,Y) | dom [alph]a = X} and I(X,Y) = {[alpha] [is an element of] P(X, Y) |[alpha] is 1-1}. For [theta] [is an element of] P(Y, X), let (P(X, Y), [theta]) denote the semigroup (P(X, Y),*) where [alpha]* [beta] = [alpha] [theta] [beta] for all [alpha], [beta] [is an element of] P(X, Y). The semigroups (T(X, Y),[theta]) with [theta] [is an element of] T(Y, X) and (I(X, Y),[theta]) with [theta] [is an element of] I(Y, X) are defined similarly. For a poset X, let OP(X), OT(X) and OI(X) denote the order-preserving partial transformation semigroup on X, the full order-preserving transformation semigroup on X and the order-preserving 1-1 partial transformation semigroup on X, respectively. For any posets X and Y, let OP(X, Y) ={[alpha] [is an element of] P(X, Y) |[alpha] is order-preserving}. For [theta] [is an element of] OP(Y, X), let (OP(X,Y), [theta]) denote the semigroup (OP(X, Y),*) where the operation * is defined as above. The semigroups (OT(X,Y), [theta]) with [theta] [is an element of] OT(Y, X) and (OI(X, Y), [theta]) with [theta] [is an element of] OI(Y, X) are defined similarly. The following facts are known. If X is a chain, then OP(X) and OI(X) are regular semigroups. For any nonempty subsets X of Z, OT(X) is regular. Moreover, for a nonempty interval X of IR, OT(X) is regular if and only if X is closed and bounded. In this research, the first known fact mentioned above is used to characterize when the semigroup (OP(X, Y), [theta]) with [theta] [is an element of] OP(Y, X) and the semigroup (OI(X, Y),[theta]) with [theta] [is an element of] OI(Y, X) are regular where X and Y are chains. It is shown that being an order-isomorphism of theta is mainly necessary and sufficient for regularity of these semigroups. We also characterize when the semigroup (OT(X,Y), [theta]) with [theta] [is an element of] OT(Y, X) is regular where X and Y are chains. This characterization is given in terms of regularity of OT(X), |, | and [theta]. Due to the above second and third known results, the characterizations of regularity of (OT(X, Y), theta) when both X and Y are nontrivial subsets of Z and when both X and Y are nontrivial intervals of IR can be given respectively in term of 0 and in terms of X and 0. Here, a nontrivial set means a set containing more than one element. Moreover, some interesting isomorphism theorems are provided where X and Y are chains. Necessary and sufficient conditions are given for that (OS(X, Y), [theta]) is equivalent to OS(X) and for that (OS(X, Y), theta) is equivalent OS(Y) where OS(X, Y) is OP(X, Y), OT(X,Y) or OI(X, Y) and [theta] [is an element of] OS(Y, X).