Construction of Semigroups with some Exotic Properties.
by
Ilya Ivanov
Moscow State University
Coauthors: Alexei Belov

There exist semigroups, groups and rings having some characteristics which are used to be considered as exotic. Namely, there exist semigroups with non-integer Gelfand-Kirillov dimension, non-nilpotent nilsemigroups and nilrings, finitely generated infinite periodic groups, and so on.

Most of these exotic objects were originally introduced either by means of infinite sets of identities or in terms of infinite sets of defining relations. So it seems to be very interesting to find finitely presented objects with similar exotic properties. An example of finitely presented associative algebra of an intermediate growth is due to V. A. Ufnarovsky, as well as the results of G. Higman, G. P. Kukin, V. Ya. Belyaev dealt with embedding of recursive presented objects (groups, associative algebras, Lie algebras) into finitely presented (we refere to [1], [2], [4], [5], [6]).

The following theorems become an object of interest in the present discussion. Theorems 1 and 2 are about the construction a finitely presented semigroups with non-integer Gelfand-Kirillov dimension and Theorem 3 is about the construction of a finitely presented semigroup G with following properties:

i) there exist a non-nilpotent ideal I=LS, where L in G;

ii) if A in G has the form A=XYYZ, there X,  Y,  Z in G then LA=0.

References

[1] Ufnarovsky V.  A On the algebras growth. (russian) Vestnik MGU. vol 1, 1978, 4, 59-65.

[2] Ufnarovsky V. A. Combinatorical and assimptotical methods in algebra. (russian) Results of science and tech. Vol. Modern math. problems. Moscow. : VINITI, 1990, 57, 5-177.

[3] Krause G. R., Lenagan T. H. Growth of algebras and Gelfand-Kirillov dimension. London: Pitman Adv. Publ. Program, 1985, 182.

[4]Kukin G.  P. The variety of all rings has Higman's property. Algebra and Analysis. Irkutsk. 1989 91-101

[5] Bokut L.  A., Kukin G. P. Algoritmic and combinatorial aldebra. Math. and its appl. 255, Kluwer Academic Publishers Group, Dordrecht, 1994. xvi+384 pp

[6] Belyaev V. Ya. Imbeddability of recursively defined inverse semigroups in finitely presented semigroups. Sibirsk. Math. Journal 25 no. 2., 1984. 50-54.

Date received: January 30, 2003