Bar-Ilan Algebra Seminar
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***  NOTE UNUSUAL TIME ***

Date: Wednesday, 3 Adar Sheni 5771 // March 9, 2011
Time: 10:00--12:00
Place: Third floor seminar room, Mathematics building

Speaker: Dr. Ilya Ivanov-Pogodaev (Moscow State University)

10:00  Construction of finitely-presented semigroups with non-integer
Gelfand-Kirillov dimension

11:00  Algebras with finite Grobner basis but algorithmically unsolvable
zero-divisors problem

Abstracts:

Construction of finitely-presented semigroups with non-integer
Gelfand-Kirillov dimension

Examples of semigroups with arbitrary Gelfand-Kirillov dimension
$\gamma>2$ are well known. However, all such semigroups with
non-integer Gelfand-Kirillov dimension are not finitely presented,
i.e. have infinite set of defining relations.
We construct finitely-presented semigroups with non-integer
Gelfand-Kirillov dimension $\gamma>2$ for a large class of recursive
numbers.

Algebras with finite Grobner basis but algorithmically unsolvable
zero-divisors problem

We construct an algebra $A$ presented by a set of relations with finite
Gr\"oebner basis such that the following problems are algorithmically
unsolvable.

Problem 1. Given an element $a\in A$, does there exist $b\in A$ such
that $ab=0$.

Problem 2. Given an element $a\in A$, does there exist $n\in N$ such
that $a^n=0$.

Note that   in the case of finite  Gr\"oebner basis the equality problem
is effectively algorithmically solvable.  Note that for finitely
presented monomial algebras these problems are
algorithmically solvable.
The construction is based on the Minsky Machine.

---------------------------------------------------------
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il>
Announcement from: Michael Schein   <mschein@math.biu.ac.il>