Finite semigroups generated by two idempotents and realized by triangular mappings (Q2901599)

The author of the article pays attention to the possibility of two-dimensional generalizations of the well known Sharkovskii theorem (1964). If we consider the set of all continuous mappings of the interval into itself and the problem of exact semigroup representations then the description of all semigroups which have exact representation with such mappings is semigroups of the form \(\langle a: a^n=a^{n+1}\rangle\) and \(\langle a: a^n=a^{n+2}\rangle\) and it is a complete description. This result can be easily proven with the Sharkovskii theorem using the ideas of its proof. In the proof of the main result of the work under review the author uses his previous result which says that if we consider the problem of the description of the finite semigroup generated with two idempotents and has exact representation by continuous interval mapping then the condition of finiteness of cyclic semigroups generated with generators is the enough condition for existing the correspond exact representation. The article considers the problem of possible finite semigroups generated by two idempotents which have exact representation in the set of triangular mappings, i.e., mappings of the form \(F(x,y) = (f(x),g(x,y))\) for some continuous mappings \(f:\, [0,\, 1] \rightarrow [0,\, 1]\) and \(g:\, [0,\, 1]^2 \rightarrow [0,\, 1]\). The main result of the article is the fact that all finite semigroups which are generated with two idempotents and have exact representation in the set of triangular mappings are groups of the form \(\langle a: a^n=a^{n+1}\rangle\), \(\langle a: a^n=a^{n+2}\rangle\) and \(\langle a: a^n=a^{n+4}\rangle\). As the example article contains the triangular map which generated the cyclic group of \(4\) elements. The interest of this example is because of un existence of interval mappings which generates finite group which contains more then \(2\) elements.