Jonathan D. H. Smith

Mal’cev Varieties

Rudiments and notations.- Centrality.- Direct decompositions.- Central isotopy and cancellation.- Plain algebras and equational completeness.- Extensions and obstructions.

An introduction to quasigroups and their representations

QUASIGROUPS AND LOOPS Latin squares Equational quasigroups Conjugates Semisymmetry and homotopy Loops and piques Steiner triple systems I Moufang loops and octonions Triality Normal forms Exercises Notes MULTIPLICATION GROUPS Combinatorial multiplication groups Surjections The diagonal action Inner multiplication groups of piques Loop transversals and right quasigroups Loop transversal codes Universal multiplication groups Universal stabilizers Exercises Notes CENTRAL QUASIGROUPS Quasigroup congruences Centrality Nilpotence Central isotopy Central piques Central quasigroups Quasigroups of prime order Stability congruences No-go theorems Exercises Notes HOMOGENEOUS SPACES Quasigroup homogeneous spaces Approximate symmetry Macroscopic symmetry Regularity Lagrangean properties Exercises Notes PERMUTATION REPRESENTATIONS The category IFSQ Actions as coalgebras Irreducibility The covariety of Q-sets The Burnside algebra An example Idempotents Burnsides lemma Exercises Problems Notes CHARACTER TABLES Conjugacy classes Class functions The centralizer ring Convolution of class functions Bose-Mesner and Hecke algebras Quasigroup character tables Orthogonality relations Rank two quasigroups Entropy Exercises Problems Notes COMBINATORIAL CHARACTER THEORY Congruence lattices Quotients Fusion Induction Linear characters Exercises Problems Notes SCHEMES AND SUPERSCHEMES Sharp transitivity More no-go theorems Superschemes Superalgebras Tensor squares Relation algebras The reconstruction theorem Exercises Problems Notes PERMUTATION CHARACTERS Enveloping algebras Structure of enveloping algebras The canonical representation Commutative actions Faithful homogeneous spaces Characters of homogeneous spaces General permutation characters The Ising model Exercises Problems Notes MODULES Abelian groups and slice categories Quasigroup modules The fundamental theorem Differential calculus Representations in varieties Group representations Exercises Problems Notes APPLICATIONS OF MODULE THEORY Nonassociative powers Exponents Steiner triple systems II The Burnside problem A free commutative Moufang loop Extensions and cohomology Exercises Problems Notes ANALYTICAL CHARACTER THEORY Functions on finite quasigroups Periodic functions on groups Analytical character theory Almost periodic functions Twisted translation operators Proof of the existence theorem Exercises Problems Notes APPENDIX A: CATEGORICAL CONCEPTS Graphs and categories Natural transformations and functors Limits and colimits APPENDIX B: UNIVERSAL ALGEBRA Combinatorial universal algebra Categorical universal algebra APPENDIX C: COALGEBRAS Coalgebras and covarieties Set functors REFERENCES INDEX

Entropy and information in evolving biological systems

Integrating concepts of maintenance and of origins is essential to explaining biological diversity. The unified theory of evolution attempts to find a common theme linking production rules inherent in biological systems, explaining the origin of biological order as a manifestation of the flow of energy and the flow of information on various spatial and temporal scales, with the recognition that natural selection is an evolutionarily relevant process. Biological systems persist in space and time by transfor ming energy from one state to another in a manner that generates structures which allows the system to continue to persist. Two classes of energetic transformations allow this; heat-generating transformations, resulting in a net loss of energy from the system, and conservative transformations, changing unusable energy into states that can be stored and used subsequently. All conservative transformations in biological systems are coupled with heat-generating transformations; hence, inherent biological production, or genealogical proesses, is positively entropic. There is a self-organizing phenomenology common to genealogical phenomena, which imparts an arrow of time to biological systems. Natural selection, which by itself is time-reversible, contributes to the organization of the self-organized genealogical trajectories. The interplay of genealogical (diversity-promoting) and selective (diversity-limiting) processes produces biological order to which the primary contribution is genealogical history. Dynamic changes occuring on times scales shorter than speciation rates are microevolutionary; those occuring on time scales longer than speciation rates are macroevolutionary. Macroevolutionary processes are neither redicible to, nor autonomous from, microevolutionary processes.

Characters of Finite Quasigroups III: Quotients and Fusion

The character theory of finite quasigroups, introduced in [4] and [5], is developed further. The Quotient Theorem relates the character theory of a quasigroup to that of its homomorphic images. The Fusion Theorem relates the character theories of different quasigroup structures on the same set. Fusion geometry interprets certain character values as solutions of optimization problems. Magic rectangle conditions impose additional constraints that facilitate the computation of character tables.

On the nilpotence class of commutative Moufang loops

The nilpotence class of the free commutative Moufang loop on n generators (n > 3) is the maximum allowed by the Bruck-Slaby Theorem, namely n − 1. This is proved by setting up a presentation of an extension of the loops multiplication group as a nilpotent group of class at most 2n − 2, and then using the Macdonald-Wamsley technique of nilpotent group theory to show that this class is exactly 2n − 2.

Abstract space–times and their Lorentz groups

It has recently been discovered [A. A. Ungar, Am. J. Phys. 59, 824 (1991); 60, 815 (1992)] that the set R3c={v∈R3 : ∥v∥<c} of all relativistically admissible velocities in Euclidean three‐space R3, with a binary operation ⊕ given by relativistic velocity addition, forms a gyrogroup (R3c,⊕). The gyrogroup (R3c,⊕) reduces to the group (R3,+) in the limit c→∞, + being the prerelativistic velocity addition (that is, the ordinary vector addition in the Euclidean three‐space R3). The binary operation ⊕ in R3c is gyroassociative and gyrocommutative, as opposed to the binary operation + in R3 which is associative and commutative. In this article we extend the study of gyrogroups into that of Lorentz groups. In particular, we find that a gyrogroup must be equipped with a cocycle form in order to be extendible into a Lorentz group. We thus study gyrogroups that are equipped with a cocycle form, and their resulting Lorentz groups. Interestingly, the cocycle form needed for the extension of gyrogroups into Lorentz gr...

On the unimodality and combinatorics of Bessel numbers

The Bessel numbers are reparametrized coefficients of Bessel polynomials. The paper investigates the analogies between Stirling numbers and Bessel numbers. A generating function for the Bessel numbers is obtained, and a proof of their unimodality is given. Stirling numbers and Bessel numbers are applied to the enumeration of orbit decompositions of various powers of permutation representations.

Bisemilattices of subsemilattices

Abstract Sets of subsemilattices of semilattices are given a natural meet-distributive bisemilattice structure. These bisemilattices are decomposed into constituent semilattices and distributive lattices. Their construction furnishes a left adjoint to the forgetful functor from meet-distributive bisemilattices to semilattices. Representation theorems for meet-distributive bisemilattices follow.

Characters of finite quasigroups IV: products and superschemes

For finite loops (as for finite groups), the character table of a direct product is the tensor product of the character tables of the direct factors. This is no longer true for quasigroups. Although non-ℨ and ℨ-quasigroups may have the same character table, the character table of Q × Q determines whether a finite non-empty quasigroup Q lies in ℨ or not. A combinatorial interpretation of the tensor square of a quasigroup character table is obtained, in terms of superschemes, a higherdimensional extension of the concept of association scheme.

Characters of finite quasigroups II: induced characters

Induced characters for finite quasigroups are defined, simplifying and generalizing the usual definition for groups. The Frobenius Reciprocity Theorem and an analogue of Artins Theorem for these characters are proved. Character rings for quasigroups are examined. Induced characters are then used to build the character table of the octonion loop.