Regularity conditions on order-preserving transformation semigroups. (Q1769341)

The symbol \(P(X)\) denotes the semigroup, under composition, of all partial transformations of the set \(X\). The symbols \(\text{dom\,}\alpha\) and \(\text{ran\,}\alpha\) denote, respectively, the domain and range of a transformation \(\alpha\in P(X)\). If \((X,\leq)\) is a poset, a transformation \(\alpha\in P(X)\) is `order-preserving' if for all \(x,y\in X\), \(x\alpha\leq y\alpha\) whenever \(x\leq y\) and the semigroup, under composition, of all order preserving transformations on \(X\) is denoted by \(OP(X)\). The symbol \(T(X)\) denotes the full transformation semigroup on \(X\) and \(OT(X)\) denotes the subsemigroup of \(T(X)\) consisting of all order-preserving total transformations on \(X\). The symbols \(\mathcal H\), \(\mathcal L\), \(\mathcal R\), \(\mathcal D\), and \(\mathcal J\) denote the usual Green's relations on a semigroup. The authors investigate these relations for the semigroups \(OP(X)\) and \(OT(X)\). For example, they show that if \(X\) is a countable chain with a least element \(0\) and a greatest element, then for \(\alpha,\beta\in OP(X)\), one has \(\alpha{\mathcal L}\beta\) if and only if \(\text{ran\,}\alpha=\text{ran\,}\beta\) and \(\alpha{\mathcal R}\beta\) if and only if \(\alpha\circ\alpha^{-1}=\beta\circ\beta^{-1}\). They show that every subgroup of such a semigroup \(OP(X)\) consists of one element. They go on to show that the proper ideals of \(OP(X)\) for such a set \(X\) are of the form \(P_r=\{\alpha\in OP(X):\text{rank\,}\alpha\leq r\}\) where \(0\leq r<|X|\) together with \(Q=\{\alpha\in OP(X):\text{rank\,}\alpha<\aleph_0\}\) in the case where \(X\) is countably infinite. Let \(T_r=\{\alpha\in OT(X):\text{rank\,}\alpha\leq r\}\) where \(0\leq r<\aleph_0\). A semigroup \(S\) is said to be `unit-regular' if for each \(a\in S\), \(a=aua\) for some unit \(u\in S\) and it is `biregular' if \(a=ax^2a\) for some \(x\in S\). Finally, it is `completely regular' if for each \(a\in S\), there exists an \(x\in S\) such that \(a=axa\) and \(ax=xa\). The emphasis in the remainder of this paper is to determine when \(P_r\) and \(T_r\) satisfy these various regularity conditions in the case where \(X\) is a countable chain with a least element.