Closed θ-invariant Sets On Quasi-lattice Ordered Groups

Let (G,G₊) be a quasi-lattice ordered group, Ωbe the collection of directed and hereditarysubsets of G₊ endowed with the topology inherited from {0, 1}^G₊. In the first part of this paper,we will give some examples of quasi-lattice ordered groups, and one of which, Example 1.3, willact as a model throughout this paper. For any t ∈G₊, define Ω_t = {B ∈Ω| t ∈B}. There is ahomeomorphism θ_t from the compact Hausdorff space Ωonto Ω_t. Given any H ∈Ω, let S(H) bethe closed θ-invariant subset generated by H. In this paper, we will clarify the concrete structureof S(H), see Theorem 3.2.Further, Proposition 3.4 of this paper paves the way for the study of the structure of θ-invariantsubset generated by a single point set of a product topological space. Several classical examplesare given to illustrate how these theorems, propositions, and corollaries work.M. Laca. studied the smallest closed θ-invariant subsets of Ωin 1999. This work triggersour interest in its corresponding concept: the largest closed θ-invariant proper subsets of Ω. Thepaper [3] gives us a hint by showing a suf?cient and necessary condition under which the largestclosed θ-invariant proper subset exists, see Lemma 4.1. We deepened and expanded such a resultby investigating conditions under which S(H) ≠Ω.Last but not least, while (G,G₊) happens to be an ordered group, the paper gets the conclu-sion that Ωexists a largest closed θ-invariant proper subset if and only if there exists a smallestsemigroup of G strictly containing G₊.