Fatemah Ayatollah Zadeh Shirazi

On Possible heights in special collections of transformation groups‎‎.

‎‎‎In ‎(topological) transformation ‎group‎ ‎‎(G,X)‎‎‏, ‎let’s ‎call‎ ‎‎h(G,X)=\sup\{n\geq0:‎ ‎there ‎exist‎ ‎‎Z_0\subsetneq\cdots\subsetneq Z_n=X‎‎ such ‎that‎ ‎‎‏Z_i‎‎s are closed invariant subsets of ‎‎‏X‏\}‎ height of transformation group ‎‎(G,X)‎‎‏ ‎(introduced ‎in ‎‎[4]).‎ I‎t is known that if h(G,X)<+\infty‎‎‏, then‎ ‎‎h(G,X)=card\{‎\overline{Gx}:x\in ‎X\}‎‎‏ ‎(see [2]). Height approach is a tool to classification of transformation groups,‎ e.g. it is evident that ‎‎h(G,X)=0‎‎‏ if and only if ‎‎(G,X)‎ is minimal. One may use this tool to introduce other parameters to classification of transformation groups like: indicator sequence and indicator topology (see [2]).

For special collections of transformation groups one may pay attention to the collection of their heights. ‎ In this talk we will pay attention on works have been done in this area (like graph transformation groups [3],‎ Alexandroff square transformation groups [1], …), also this talk will be motivated by diagrams and discussing on‎ possible heights of other collections of transformation groups too.‏

[1] F. Ayatollah Zadeh Shirazi, F. Ebrahimifar, R. Yaghmaeian, H. Yahyaoghli, Possible Heights of Alexandroff Square Transformation Groups , Punjab University Journal of Mathematics, 2020 (52/6), 19—29‎

[2] F. Ayatollah Zadeh Shirazi, N. Golestani, On classifications of transformation semigroups: indicator sequences and indicator topological spaces , Filomat, 2012 (26:2), 313—329‎

[3] F. Ayatollah Zadeh Shirazi, A. Hosseini, Z. Nili Ahmadabadi, Possible heights of graph transformation groups , Journal of Mathematical Analysis, 2021 (Volume 12, Issue 4), 1—9

[4] M. Sabbaghan, F. Ayatollah Zadeh Shirazi, Transformation semigroups and transformed dimensions , Journal of Sciences, Islamic Republic of Iran, 2001 (12, no.1), 65—75‎