




Universal Algebra / Algèbre universelle (J. Hyndman and S. Wismath, Organizers)
 JI YOUNG CHOI, Iowa State
The dimension of a finite algebra

Let A be a finite algebra, of size n. Let V be the
variety generated by A. Then the dimension of A is the
smallest power d of A such that A^{d} contains a subalgebra that
is free in the variety V. Note that d is welldefined,
since it is at most equal to n. The concept originates in the theory
of permutation groups, where it is also known as the ``minimal base
size''. We discuss the problem of determining the dimension of a given
finite algebra.
 G. GRATZER, Department of Mathematics, University of
Manitoba, Winnipeg, Manitoba R3T 2N2
The case of the shrinking proof: The Dilworth Theorem on embedding
every lattice into a uniquely complemented lattice

Let L be a lattice, and let a < b < c be elements of L. We
adjoin freely a relative complement u of b in [a,c] to L to
form the lattice K. For two polynomials A and B over L È{u}, we find a very simple set of conditions under which A and B
represent the same element in K. It follows readily that in K all
pairs of relative complements in [a, c] can be described.
As an immediate consequence, we get the classical result of
R. P. Dilworth: Every lattice can be embedded into a uniquely
complemented lattice. We also get the stronger form due to C. C. Chen
and G. Grätzer: Every at most uniquely complemented bounded lattice
has a {0,1}embedding into a uniquely complemented lattice. Some
stronger forms of these result are also presented.
The proof by Chen and Grätzer cut roughly in half the proof of
Dilworth; the present approach cuts roughly in half the proof of Chen
and Grätzer. Maybe, someday, there will be a really short proof that
can be presented as a footnote?
 B. LAROSE, Concordia University and Champlain Regional College
Constraint satisfaction problems, posets and Taylor operations

Let T be a finite relational structure. Consider the following
decision problem: given a structure S of the same type as T, is
there a homomorphism from S to T? It has been conjectured that
every such problem is either tractable or NPcomplete, and that the
dichotomy is related to the existence of certain idempotent operations
preserving the basic relations of the target structure. We present a
short survey of recent results in this area, as well as some related
results on retraction and extendibility problems for finite posets.
(Joint work with A. Krokhin)
 JB NATION, University of Hawaii, Honolulu, Hawaii 96822, USA
A gluing construction for projective planes

A projective plane is a simple, complemented, modular lattice of height
3. We describe a gluing construction which can sometimes be used to
make a plane of order n from planes of smaller order. A key step in
the construction involves finding directed graphs which are a
generalization of symmetric (v,k,l) block designs. These
graphs are interesting in their own right.
 B. SANDS, University of Calgary, Calgary, Alberta T2N 1N4
Finite distributive lattices and the splitting property

Given a maximal antichain A of an ordered set P, say that
A splits if there is a partition A=BÈC such that
P=B ÈC¯, where
B={x Î P:x ³ b for some b Î B} and C¯ = {x Î P:x £ c for some c Î C}. 

Say that P has the splitting property if every maximal antichain
of P splits. Ahlswede, Erdös and Graham introduced these notions in
1995 and proved that every finite Boolean lattice has the splitting property.
We characterize all finite distributive lattices with the splitting
property. It turns out that there are exactly three others
besides Boolean lattices.
We also introduce the notion of splitting number, which measures
how close a finite ordered set is to being splitting, give what we know
of this concept, and state some interesting problems. We then
concentrate on specific classes of distributive lattices, socalled
``grids'' obtained as the product of two chains.
 M. VALERIOTE, McMaster University, Hamilton, Ontario L8S 4K1
Varieties with few finite members

The Gspectrum of a variety V is the function g_{V}: N® N, where g_{V}(n) is equal to the number of ngenerated
members of V, up to isomorphism. The study of Gspectra was
initiated by Berman and Idziak in the 1990s.
In my talk I will discuss a result with Idziak and McKenzie on finitely
generated varieties whose Gspectra are polynomially bounded. It
turns out that these varieties have very nice structure.
 R. WILLARD, University of Waterloo, Waterloo, Ontario N2L 3G1
Failure of injectivity in finitely generated quasivarieties

In 1997 I introduced an ordinalvalued function measuring the degree to
which a finite algebra fails to be injective in the quasivariety it
generates. (The motivation for such a function comes from the
``natural duality theory'' of D. Clark and B. Davey.) In this talk I
will describe the progress that J. Hyndman and I are making in our
attempt to understand this function within the context of unary
algebras.

