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Title:
On Kirby C. Smith's Proof That
the Lattice of Left Ideals in a Finite Centralizer Near-Ring Is
Distributive
Speaker: Jonathan Farley
Time: 10:30 - 12:00, 27/8/09
Room: KG 712
Abstract:
Let G be a finite group. Let A be a subgroup of the group of
automorphisms
of G. Let M_A(G) denote the (right) centralizer near-ring of G with
respect
to A.
In [Smith 1982, Theorem 3], Smith purports to prove that the lattice of
left
ideals is distributive. Ecker [Ecker 1997] has written, under the
heading,
"What we intend to prove---Questions in classical near-ring theory,"
"Does every near-ring M_{Aut(G)}(G) have a distributive lattice of
left
ideals? This is an open question; existing 'proofs' are known to have
gaps,
therefore it might be reasonable to look for a counterexample."
Although it happens rarely, sometimes information on the internet is
inaccurate.
Juergen Ecker, "Computing with Near-Rings---Algorithms and
Implementation"
[Internet]. Available from:
http://www.algebra.uni-linz.ac.at/Projects/FWF/P11486-TEC/node7.html[accessed
August 6, 2009].
Kirby C. Smith, "The Lattice of Left Ideals in a Centralizer Near-Ring
Is
Distributive," Proceedings of the American Mathematical Society 85
(1982),
313--317.
Title:
Between Q and Stone: some aspects of quasi-Stone algebras
Speaker: Sara-Kaja Fischer, (University of Bern)
Time: 10:30 - 12:00, 21/7/09
Room: KG 712
Abstract:
A "quantifier" is a certain type of join-preserving closure operator on
a lattice. The notion of a quantifier on a Boolean algebra was
introduced in the 1950's by Halmos, and later considered by several authors in
different contexts. Cignoli, in 1991, investigated quantifiers on distributive lattices
as well as the corresponding Priestley spaces. A bounded
distributive lattice with a quantifier is called a Q-distributive lattice and its dual
space a Q-space.
In 1993, Sankappanavar and Sankappanavar generalised the concept of Stone
algebras by defining and investigating the class of "quasi-Stone algebras".
A first connection between quasi-Stone algebras and Q-distributive lattices was
described by Gaitan in 2000. On the basis of Cignoli\u2019s Q-spaces, Gaitan found a description of the Priestley spaces
of quasi-Stone algebras, the so-called QS-spaces. He stated as well that quasi-
Stone algebras may be identified with Q-distributive lattices satisfying a certain
condition.
We study amalgamation in the variety of quasi-Stone algebras, in an attempt to solve some of the problems from the 1993 paper of Sankappanavar and Sankappanavar.
R. Cignoli. Quantifiers on distributive lattices. Discrete Math., 96:183-197,
1991.
H. Gaitan. Priestley duality for quasi-stone algebras. Studia Logica, 64:83-92,
2000.
H. Gaitan. Varieties of quasi-stone algebras. Annals of Pure and Applied
Logic, 108:229-235, 2001.
P. R. Halmos. Algebraic logic, i. monadic boolean algebras. Compositio Math.,
12:217-249, 1955.
H. A. Priestley. Stone lattices: a topological approach. Fund. Math., 84:127-143, 1974.
N. H. Sankappanavar and H. P. Sankappanavar. Quasi-stone algebras. Math.
Log. Quart., 39:255-268, 1993.
Title:
Islands on a triangular grid and in higher dimensions
Speaker: Gabriella Pluhar, (Eötvös University, Budapest)
Time: 10:15 - 11:45, 10/3/09
Room: MZ 003 B
Abstract:
For each unit square of a square lattice a real number is given, its
height. A rectangle on the lattice is called a rectangular island iff
the height of every unit square of the rectangle is bigger than the height
of the adjacent unit squares. In other words, if there exists a possible
water-level by which the rectangle is an island in the usual sense.
Gabor Czedli has proved that the maximum of the number of rectangular islands on an m x n square lattice is [(mn+m+n-1)/2].
There are other cases when we assign numbers to cells; for
example, a number may mean a colour on a gray-scale (before we convert the picture to
black and white), transparency (against X-rays) or melting temperature.
We give lower and upper estimates for the maximum of the number of
islands in higher dimensions and on the triangular grid. For the upper
bounds we use pure algebraic results of Gabor Czedli, Andras
Huhn and Tamas Schmidt.
Title:
Combinatorics of words over groups and semigroups
Speaker: Csaba Szabo, (Eötvös University, Budapest)
Time: 13:45 - 15:15, 9/3/09
Room: BA 9912
Abstract:
The equivalence problem asks, whether or not two expressions over an algebra A
attain the same value for every substitution. The equation solvability problem
asks, whether or not two expressions over A attain the same value for at
least one substitution. The talk consists of two
parts. In the first part we generalize the two complexity problems for groups.
In the second part we investigate terms and
polynomials for finite 0-simple semigroups. These semigroups are
building blocks for semigroups, as simple groups are for groups. We describe
the structure of these semigroups and the equivalence and solvability
problems for them.
Title:
Complexity of Equivalence and Solvability Problems over Finite Algebras
Speaker: Gabor Horvath (University of Hertfordshire, UK)
Time: 10:15 - 11:45, 17/2/09
Room: HS 12
Abstract:
The talk is about determining the complexity of checking an identity or
determining if an equation has a solution over a given finite algebra.
A finite algebra A is given. The equivalence problem asks,
whether or not two expressions over A attain the same value for
every substitution.
The equation solvability problem asks, whether or not two expressions
over A attain the same value for at least one substitution.
These questions arise naturally in several areas of algebra. For rings
the equivalence problem asks, whether or not a polynomial is
identically 0 over the ring. The equation solvability problem asks,
whether or not a polynomial has a root over the ring.
We show some useful methods one can attack these problems for
classical algebraic structures. These raise several new questions,
the so-called extended problems. We define the extended equivalence
and extended solvability problems and show the existing results.
Title:
On a new kind of homogeneity
Speaker: Dragan Masulovic
Time: 10:15 - 11:45, 9/12/08
Room: T 111
Abstract:
A structure is called homogeneous if every isomorphism between finite
substructures of the structure extends to an automorphism of the structure.
Recently, P. J. Cameron and J. Ne\v{s}et\v{r}il introduced a relaxed version
of homogeneity: we say that a structure is homomorphism-homogeneous if every
homomorphism between finite substructures of the structure extends to an
endomorphism of the structure.
Not much is known about homomorphism-homogeneous structures.
Homomorphism-homogeneous posets were characterized in 2007 by D.
Ma\v{s}ulovi\'c and the characterization of countable posets with respect to
various types of morphisms can be found in a recent paper by P. J. cameron
and D. Lockett (accepted for publication). Moreover, finite
homomorphism-homogeneous tournaments (with loops) were characterized in 2008
by A. Ili\'c, D. Ma\v{s}ulovi\'c and U. Rajkovi\'c.
In this talk we shall survey some of these results and
discuss possibilities for further investigations.
Title:
The Wide Partition Conjecture: A Survey
Speaker: Jonathan Farley
Time: 10:15 - 11:45, 11/11/08
Room: T 950
Abstract:
Consider a partition $\lambda=(\lambda_1, \lambda_2, \dots)$ of a positive integer $n$, viewed as a Young diagram: row $i$ has $\lambda_i$ boxes. Your task is to fill in each box with a number---for row $i$ you are only allowed to use the numbers 1, 2, 3, ..., up to $\lambda_i$---so that no number appears twice in the same row or column.
For some shapes $\lambda$ you can do this, for some you cannot. The Wide Partition Conjecture asserts for what shapes this can, and for what shapes this cannot, be done.
In this talk we survey what is known (assuming certain books were mailed from the United States to Austria and not to Australia) and present some ideas for how to tackle the conjecture. This talk will be very low on new results and is IDEAL FOR UNDERGRADUATES (as well as everyone else).
Title:
How to Build the Perfect Terrorist Cell
Speaker: Jonathan Farley
Time: 10:15 - 11:45, 21/10/08
Room: T 950
Abstract:
Consider the class of connected n-element posets with M maximal elements such that every node has at most s lower covers. What poset in this class has the fewest k-element cutsets?
Links:
Terror and Beauty - The European Institute for Mathematical Methods in Counterterrorism.
Toward a Mathematical Theory of Counterterrorism.
Title:
Digraph coloring problems: special triads
Speaker: Todd Niven (Praha)
Time: 13:00 - 14:00, 30/11/07
Room: KG 712
Abstract:
A digraph colouring problem for a fixed finite digraph H is
the following:
INSTANCE: A finite digraph G.
PROBLEM: Does there exists a digraph homomorphism from G to H?
In this talk we present a recent result on the computational complexity
of oriented tree colouring problems. Specifically, we classify the
complexity of special triad colouring problems and show that they are
either solvable in polynomial time, or NP-complete. Our result
significantly generalises previous work by Hell, Nesetril and Zhu and
provides the smallest known example of an NP-complete oriented tree.
Title:
The algebraic approach to the constraint satisfaction problem: recent progress
Speaker: Libor Barto (Praha)
Time: 11:15 - 12:15, 30/11/07
Room: BA 9909
Abstract:
Many decision problems in combinatorics, computer science,
artificial intelligence, logic, etc. can be expressed as so called
Constraint Satisfaction Problems (CSPs). For a fixed relational
structure R, CSP(R) is the following decision problem:
INPUT: A relational structure S of the same signature as R.
OUTPUT: Is there a homomorphism from S to R?
The central problem in this area is the Dichotomy Conjecture of Feder
and Vardi stating that, for any R, CSP(R) is either solvable in
polynomial time or NP-complete. I will talk about the universal
algebraic approach to this problem and mention some recent developments.
Title:
Automated reasoning in algebra
Speaker: David Stanovsky (Praha)
Time: 16:00 - 17:00, 29/11/07
Room: T 911F
Abstract:
Recent development in automated resoning, together with
constant increase of computer power, start to make automated theorem
provers useful in algebraic research - at least in some fields. I will
present a couple of pioneering works and some results that, in my
opinion, may show future directions.
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