The authors define three concepts of ideal extension properties of a topological semigroup \(S:S\) has the ideal extension property (IEP) if, for each closed subsemigroup \(T\) of \(S\) and each closed ideal \(I\) of \(T\), there is a closed ideal \(J\) of \(S\) such that \(J\cap T=I\). \(S\) has the weak ideal extension property (WIEP) if there is an ideal \(J\) (not necessarily closed) of \(S\) extending each closed ideal of each closed subsemigroup. \(S\) has the feeble ideal extension property (FIEP) if, for each closed subsemigroup \(T\) of \(S\) and each closed ideal \(I\) of \(T\), there is a congruence \(\sigma\) (not necessarily closed) extending the Rees congruence \(\Delta_T\cup(I\times I)\) on \(T\) (i.e., \(\sigma\cap(T\times T)=\Delta_T\cup(I\times I))\). Let \(S\) be a topological semigroup and let \(a\in S\). Let \(\theta(a)=\{a^n:n\in Z^+\}\). Then \(\Gamma(a)=\overline{\theta(a)}\) is called the monothetic subsemigroup of \(S\) generated by \(a\). If \(S=\Gamma(a)\) for some \(a\in S\) then \(S\) is called a monothetic semigroup. A topological semigroup \(S\) is called \(\Gamma\)-compact if each of its monothetic subsemigroups is compact. The main theorem of this paper is the following: If \(S\) is a \(\Gamma\)-compact regular semigroup then the following four conditions are equivalent: (i) \(S\) has monothetic index 1. (ii) \(S\) is completely regular. (iii) \(S\) satisfies WIEP. (iv) \(S\) satisfies FIEP.