Sokolovskaya, A. M. A method for constructing semilattices of \(G\)-compactifications. (English. Russian original) Zbl 1161.54012 Math. Notes 82, No. 6, 827-835 (2007); translation from Mat. Zametki 82, No. 6, 916-925 (2007). 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. Reviewer: Vitalij Chatyrko (Linköping) MSC: 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) Keywords:\(G\)-Tychonoff space; semilattice of G-compactifications; equipartition; semilattice of equipartitions PDFBibTeX XMLCite \textit{A. M. Sokolovskaya}, Math. Notes 82, No. 6, 827--835 (2007; Zbl 1161.54012); translation from Mat. Zametki 82, No. 6, 916--925 (2007) Full Text: DOI References: [1] K. L. Kozlov and V. A. Chatyrko, ”On G-compactifications,” Mat. Zametki 78(5), 695–709 (2005) [Math. Notes 78 (5–6), 649–661 (2005)]. · Zbl 1114.54016 [2] J. M. Smirnov and L. N. Stojanov, ”On minimal equivariant compact extensions,” C. R. Acad. Bulgare Sci. 36(6), 733–736 (1983). · Zbl 0522.54027 [3] Yu. M. Smirnov, ”Can simple geometrical objects be maximal compact extensions for ĭ n ?” Usp. Mat. Nauk 49(6), 213–214 (1994) [Russ. Math. Surv. 49 (6), 214–215 (1994)]. · Zbl 0890.54022 [4] Yu. M. Smirnov, ”Minimal topologies on acting groups,” Usp. Mat. Nauk 50(6), 217–218 (1995) [Russ. Math. Surv. 50 (6), 1308–1310 (1995)]. · Zbl 0888.54040 [5] A. M. Sokolovskaya, ”The existence of G-Tychonoff spaces whose semilattices of G-compactifications have minimal but not least elements,” in Aleksandrov’s Readings–2006. An International Conference Dedicated to the 110th birthday of Pavel Sergeevich Aleksandrov, Moscow, Russia, 2006 (Moscow, 2006) pp. 37–38. [6] J. de Vries, ”On the existence of G-compactifications,” Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26(3), 275–280 (1978). · Zbl 0378.54028 [7] J. de Vries, Topological Transformation Groups I. A Categorical Approach (Mathematisch Centrum, Amsterdam, 1975). · Zbl 0315.54002 [8] G. Birkhoff, Lattice Theory, 3rd ed. (Amer. Math. Soc., Providence, RI, 1967; Nauka, Moscow, 1984). [9] A. G. Kurosh, Group Theory (Nauka, Moscow, 1967) [in Russian]. [10] R. Engelking, General Topology, 2nd ed. (Mir, Moscow, 1986; Heldermann, Berlin, 1989). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.