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Universal Algebra and Lattice Theory / Algèbre universelle et théorie des treillis
Org: Jennifer Hyndman (UNBC)

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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 ad-hoc 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 element-wise 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 first-order 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 so-called 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 well-studied model-theoretic 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 union-closed set systems

The union-closed 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 semi-simple 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 semi-simple 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  37240-0001, 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, non-isomorphic 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. Gian-Carlo 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), 694-711) 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.