A method for constructing semilattices of \(G\)-compactifications (Q941921)

An equipartition of the pair \((X,Y)\) of a \(G\)-space \(Y\) and its invariant subset \(X\) is a decomposition of \(Y\) such that each point of \(Y \setminus X\) is an element of the decomposition, the quotient mapping is perfect and an action by \(G\) on the quotient space can be defined compatibly with the action on \(Y\). The set of all equipartitions is a complete upper semilattice (which coincides with the semilatice of \(G\)-compactifications of some (pseudocompact) \(G\)-space \(P\) when \(Y\) is compact). Let now \(Z_i\) be a compact \(G\)-space, \(T_i\) its invariant subspace and \(t_i \in T_i\), where \(i = 0,1\). Denote by \(H_0\) the stabilizer of the point \(t_0\). The author shows that if the restrictions of the actions by \(G\) on the sets \(T_0, T_1\) are transitive and each point \(t\in T_0\setminus t_0\) is a cluster point of the set \(H_0 t\) then there exists a compact \(G\)-space \(Y\) and its invariant subset \(X\) such that the semilattice of equipartitions of the pair \((X,Y)\) is the union of the semilattices of equipartitions of the pairs \((T_0, Z_0)\) and \((T_1, Z_1)\) with the identification of their maximal elements. This construction allows to present for each integer \(n \geq 2\) examples of (pseudocompact) \(G\)-spaces such that the semilattices of their \(G\)-compactifications have \(n\) minimal elements but not a smallest one. Recall that for the semilattices of ordinary compactifications a minimal element exists iff the smallest element exists and in that case the space is locally compact.