
Combinatorics/Algebra Seminar Spring 2014 Archive:

April 28

SungYell Song: On relation graphs of 3class nonsymmetric association schemes.

April 14, 21

Cliff Bergman: Verbose rings.
Abstract:
An ideal I of a ring R is called fully invariant if I is
preserved by every endomorphism of R. For a subset S of
Z< x_{1}, x_{2}, . . . >,
the ideal I is the verbal ideal induced by S if I is generated by
{ p(a_{1}, . . . , a_{n})
: p in S, a_{1}, . . . , a_{n} in R }.
It is easy to show that every verbal ideal is
fully invariant. The converse is false in general. We shall show that if
R satisfies x^{k} = x
for some k > 1, then every fully invariant ideal of R is verbal. The proof is a nice application of natural duality as developed by Davey and Werner.

April 7

William DeMeo (University of South Carolina): From subgroup lattice intervals to group properties.
Abstract:
We explain how to use the shape of an interval in a subgroup lattice to deduce properties of the underlying group. We describe a few properties that are "interval enforceable" in this sense, and a few that are not, and we connect these ideas to an old open problem in universal algebra.

March 31

Jonathan Smith: Report from the AMS Spring Meeting, Knoxville, TN.

March 17, 24

Spring break (and aftermath): no seminar.

February 24

Anna Romanowska (Warsaw University of Technology): Duality for some classes of convex sets.
Abstract:
There is a well known selfduality for the category of
finitedimensional (real) vector spaces which can be adjusted to the
category of corresponding affine spaces. This duality can be extended to
the category of all affine spaces. However, it cannot be restricted to
provide a duality for the category of convex subsets of (real) affine
spaces. We will discuss the problem of duality for some classes of convex
sets. We consider convex subsets of real affine spaces as abstract
algebras, socalled barycentric algebras. Dualities for some classes of
barycentric algebras will be described. In constructing some of these
dualities, we use methods of dualizing socalled Płonka sums of algebras.

February 17, March 3, 10

Jonathan Smith: Quantum quasigroups, I, II, III.
Abstract:
Quantum quasigroups (and quantum loops) are nonassociative extensions of Hopf algebras, with a much simpler axiomatization. They include previous attempts at extending Hopf algebra structure to nonassociative situations, in particular the MoufangHopf algebras of Benkart et al., and the Hbialgebras of PérezIzquierdo. We show that Hopf algebras form quantum quasigroups, and provide sufficient conditions for a quantum quasigroup to augment to a Hopf algebra. A key example of a quantum quasigroup provides an algebra of rooted binary trees, serving as indices for nonassociative powers.

January 27; February 3, 10

Candidate colloquia: no seminar.

January 20

King Holiday: no seminar.
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