Around P-small subsets of groups - Protasov, I.V.; Protasova, K.D. (Carpathian Mathematical Publications, 2014)
A subset X of a group G is called P-small (almost P-small) if there exists an injective sequence (g_n)_nω in G such that the subsets (g_nX)_nω are pairwise disjoint (g_nXg_mX is finite for all distinct n,m), and weakly P-small if, for every nω, there exist g_0, … ,g_nG such that the subsets g_0 X, ..., g_n X are pairwise disjoint. We generalize these notions and say that X is near P-small if, for every nω, there exist g_0, … ,g_nG such that g_iXg_jX is finite for all distinct i,j {0,…, n}. We study the relationships between near P-small subsets and known types of subsets of a group, and the behavior of near P-small subsets under the action of the combinatorial derivation and its inverse mapping.