


Universal Algebra and Lattice Theory / Algèbre universelle et théorie des treillis Org: Jennifer Hyndman (UNBC)
 ERIN BEVERIDGE, University of Northern British Columbia, Prince George,
British Columbia
Irresponsible homomorphisms and rank infinity

Ross Willard has shown that rank can be used as a tool for determining
if an algebra is strongly dualizable. In particular, if a dualizable
algebra has finite rank then we know that the algebra is strongly
dualizable. Determining if an algebra has rank infinity has previously
been an adhoc process. We describe a process for showing that an
algebra has rank infinity by introducing the concept of lifting dense
sets and irresponsible homomorphisms. We demonstrate this process on
an example family of algebras, ultimately showing that each algebra in
the family is dualizable but not strongly dualizable.
 DAVID CASPERSON, University of Northern British Columbia, Prince George,
British Columbia V2N 4Z9
ALDOR and universal algebra

Many computer algebra packages and languages, for instance,
MATHEMATICA, GAP, or MAPLE, are clearly oriented
towards classical algebra, and in particular to ring theory and group
theory. Because the underlying universes are often infinite, they
focus on elementwise computations. Although these packages are
capable of computing Gröbner bases, they have no generic
representation of congruences or quotient algebras.
The ALDOR programming language (freely available from
www.aldor.org) is also designed for algebraic computation.
Although the ALDOR packages developed to date focus on classical
algebra, the language is designed in such a way that it is directly
able to represent constructs commonly used in universal algebra. In
particular, the fact that types are first class objects in ALDOR
makes it possible to directly express notions such as product,
congruence, and reduct.
I shall discuss why ALDOR is a good language for writing
algorithms and performing calculations of a universal algebraic nature,
and illustrate the use of ALDOR by performing calculations on
small unary algebras.
 DEJAN DELIC, Ryerson University, Toronto, Ontario
Hereditarily homogeneous locally finite groups

One of the outstanding problems in the contemporary general algebra has
been the quest for the description of locally finite equational
theories (i.e. locally finite classes of algebras defined via
equalities of terms in a given language) whose firstorder theory is
decidable. Due to the profound work done in the field by R. McKenzie
and M. Valeriote, the problem has been reduced to understanding the
structural features of such classes of modules over finite rings and
socalled discriminator equational theories. A discriminator
equational theory, loosely speaking, is one equivalent to a class of
sheaves of continuous functions over algebras endowed with a Boolean
topology.
Over the past few years the connection between the decidability of
discriminator equational theories and the wellstudied modeltheoretic
notion of homogeneity has become more transparent. In particular, the
need for the characterization of hereditarily homogeneous locally
finite algebras (i.e. the locally finite algebras all of whose
finite subalgebras are homogeneous) has come into focus. In this talk,
we shall attempt to highlight the main features of such groups as well
as present some recent results for discriminator equational theories
arising from various classes of locally finite solvable groups.
 DWIGHT DUFFUS, Emory University, Atlanta, Georgia 30322, USA
A lattice theoretic approach to unionclosed set systems

The unionclosed set system conjecture, which appears to have been
formulated by Peter Frankl in the 1970's, is that any nontrivial such
system has an element in at least half the sets. There is an easy
translation, made independently by several researchers, into statements
about lattices. Attempts to solve the conjecture in its lattice form
have led, not surprisingly, to partial results for special classes of
lattices. The lattice theory viewpoint also suggests several
approaches to the problem and some interesting variants on the original
conjecture. In this talk, partial results and related conjectures,
obtained with Bill Sands [University of Calgary], are presented, as
well as some experimental results, done with Bob Roth [Emory
University].
 JIE FANG, University of Los Andes, Bogota, Colombia
The endomorphism kernel Property in finite
semisimple Ockham algebras

An algebra A has the endomorphism kernel property if every congruence
on A, other than the universal congruence, is the kernel of an
endomorphism on A. Here we consider this property when A is a
finite semisimple Ockham algebra and describe the structure of these
algebras that have the property through Priestley duality.
 GEORGE GRATZER, University of Manitoba, Winnipeg, Manitoba R3T 2N2
Games sectional complements play

One of the major ways of obtaining finite lattices with a given
congruence lattice is by using chopped lattices: a partial lattice
obtained fom a finite lattice by chopping off the unit element. By a
joint result of mine with H. Lakser, the chopped lattice and its ideal
lattice (a lattice!) have the same congruence lattice.
With E. T. Schmidt, we raised the question when is the ideal lattice of
a finite sectionally complemented chopped lattice again sectionally
complemented. Our best result was that if we obtain a finite chopped
lattice by taking two finite sectionally complemented lattices and
identify the zeroes and an atom chosen from each, then the resulting
sectionally complemented chopped lattice has a sectionally complemented
ideal lattice.
With various examples we show that this result is best possible. (Joint
work with H. Lakser and M. Roddy.)
With E. T. Schmidt, in 1962, from a finite poset we constructed a
sectionally complemented chopped lattice and we proved that the ideal
lattice of this chopped lattice is again sectionally complemented. We
generalize this result to quasi ordered set (rather than posets) and we
give an interesting application, solving a problem I proposed with
E. T. Schmidt. (Joint work with H. Lakser.)
 BENOIT LAROSE, Concordia University, Montreal, Quebec
The complexity of solving polynomial equations over finite
algebras

(work in progress, with L. Zadori)
Let A be a finite algebra. We study the computational
complexity of the following decision problem: given a finite system
S of polynomial equations over A, does S admit a
solution? Dichotomy results are presented for several classes of
algebras.
 RALPH MCKENZIE, Vanderbilt University, Department of Mathematics,
Nashville, Tennessee 372400001, USA
The interplay between structural insight and algorithmic
problems in general algebra

In 1992, B. Hart, S. Starchenko and M. Valeriote proved Vaught's
conjecture for varieties. Every variety (equationally defined class of
algebras) of countable signature has either a continuum, or at most
countably many, nonisomorphic denumerable models. They proved that
every such variety with few models admits a decomposition into a
product of a combinatorial variety and an affine variety (equivalent to
a variety of all modules over some ring).
In 1986, R. McKenzie and M. Valeriote proved that every locally finite
variety with decidable first order theory admits a decomposition into a
product of a combinatorial variety, an affine variety, and a
discriminator variety. The proof afforded an algorithm which inputs a
finite algebra F, and outputs a finite ring R
with unit, so that the variety W generated by F
is decidable iff the theory of unital modules over R is
decidable.
In 1993, R. McKenzie proved that there is no algorithm to determine,
for a finite algebra F, whether the variety W
generated by F is residually finite. Likewise there is no
algorithm to determine if W is finitely axiomatizable.
All these results are related by dependence on the techniques (or at
least the insights) flowing from the structure theory for finite
algebraic systems called "tame congruence theory". In this talk, I
will attempt to convey the substance of this theory, and its
contribution to these results.
 BOB QUACKENBUSH, University of Manitoba, Winnipeg, Manitoba
The classical umbral calculus via universal algebra

The classical umbral calculus is a magical deductive system used to
give slick proofs of well known properties of the Bernoulli numbers and
related topics. In this context, magical is a highly pejorative term,
since the rules of the deductive system are unexplained and seemingly
inexplicable. GianCarlo Rota and his school have given a sound
mathematical foundation and wide extention of the umbral method which
fits beautifully with Rota's linear algebraic approach to combinatorics
(and which is described in Roman's "The Umbral Calculus").
Unfortunately, the magical crispness of the classical umbral method is
lost. Rota and Taylor (The Classical Umbral Calculu. SIAM
J. Math. Anal. 25(1994), 694711) attempt to correct this, but
not to my liking. In this talk, I will give a universal algebraic
approach: choose a convenient underlying set, choose some appropriate
operations on this set and investigate how these operations interact.
 MICHAEL RODDY, Brandon University, Brandon, Manitoba R7A 6A9
Bruns' conjecture

Around 1980 the late Gunter Bruns made a conjecture about varieties of
modular ortholattices (MOL's for short). Roughly speaking, he
conjectured that every variety of MOLs which is not generated by
algebras of height two or less, must contain an orthocomplemented
projective plane (i.e. a simple algebra of height three). My Phd
thesis was a small partial confirmation of this conjecture. In the mid
90's Christian Herrmann and I published a short paper which established
a very strong form of Bruns' Conjecture for varieties generated by
atomistic algebras. I will discuss this and other recent developments
(mostly due to Herrmann and his students) in this talk.
 BOZA TASIC, University of Waterloo, Ontario
HSP ¹ SHPS for commutative rings with identity

Let I, H, S, P, P_{s} be the usual operators on classes of
rings: I and H for isomorphic and homomorphic images of rings and
S, P, P_{s} respectively for subrings, direct, and subdirect
products of rings. If K is a class of commutative rings with
identity (and in general of any kind of algebraic structures) then the
class HSP(K) is known to be the variety generated by the
class K. Although the class SHPS(K) is in
general a proper subclass of the class HSP(K) for many
familiar varieties HSP(K) = SHPS(K). Our goal is
to give an example of a class K of commutative rings with
identity such that HSP(K)\not = SHPS(K). As a
consequence we will describe the structure of two partially ordered
monoids of operators.
 MATT VALERIOTE, McMaster University, Hamilton, Ontario
Definable principal congruences and solvability

An equationally defined class of algebraic structures V is
said to have Definable Principal Congruences (DPC) if there is a first
order formula defining the principal congruences of the algebras in
V. In this talk I will present joint work with Pawel
Idziak, Keith Kearnes, and Emil Kiss that considers solvable
congruences and algebras in a DPC equational class. We find that DPC
imposes stronger centrality conditions like nilpotence, or strong
abelianness.
 SHELLY WISMATH, University of Lethbridge, Lethbridge, Alberta T1K 3M4
Inflations and generalized inflations

An inflation of an algebra B is a new algebra formed by attaching a
set S_{b} of new elements to each base element b Î B, with the
condition that elements in S_{b} always act like b in any products in
the new algebra. Inflations have been extensively studied, particularly
for semigroups, and are connected to normal identities of an algebra.
Clarke and Monzo recently introduced the idea of a generalized
inflation of a semigroup. In this talk we consider generalized
inflations of arbitrary type, and discuss some properties of inflations
and generalized inflations.

