Finite Bruck loops

Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops , showing that is essentially the direct product of a Bruck loop of odd order with a -element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite -element Bruck loops are -loops, leaving open the question of whether such obstructions actually exist.

     

Categorical Skew Lattices

Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of forbidden subalgebras. We also consider the subclass of strictly categorical skew lattices.     

Permanently oriented antibody immobilization for digoxin determination with a flow-through fluoroimmunosensor

This paper reports a new flow-through fluoroimmunosensor, the function of which is based on antibodies immobilized on an inmunoreactor of controlled-pore glass (CPG), for determination of digoxin, used in the treatment of congestive heart failure and artery disease. The immunosensor has a detection limit of 1.20 microg L(-1) and provides high reproducibility (RSD=4.5% for a concentration of 0.0025 mg L(-1), and RSD=6.7% for 0.01 mg L(-1)). The optimum working concentration range was found to be 1.2 x 10(-3)-4.0 x 10(-2) mg L(-1). The lifetime of the immunosensor was about 50 immunoassays; if stored unused its lifetime can be extended to three months. A sample speed of about 10-12 samples per hour can be attained. Possible interference from substances with structures similar to digoxin (morphine, heroin, tebaine, codeine, pentazocine and narcotine) was investigated. No cross-reactivity was seen at the highest digoxin: interferent ratio studied (1:100). The proposed fluoroimmunosensor was successfully used to determine digoxin concentrations in human serum samples.     

Ken Kunen: Algebraist

Although better known as a set theorist, Ken Kunen did important work in algebra, particularly in quasigroup theory and loop theory. This paper describes some of this work, especially his use of automated deduction tools, with a lot of attention given to projects done in collaboration with the author.     

Quadratic differential equations in -graded algebras

Quadratic differential equations whose associated algebra has an automorphism of order two are studied. Under hypotheses that naturally generalize the cases where the even or odd part of the algebra is one dimensional, the following are examined: structure theory of the associated algebra (ideal structure, simplicity, solvability, and nilpotence), derivations and first integrals, trajectories given by derivations, and Floquet decompositions.

     

Inverse semigroups with idempotent-fixing automorphisms

A celebrated result of J. Thompson says that if a finite group \(G\) has a fixed-point-free automorphism of prime order, then \(G\) is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.     

A Generalization of Moufang and Steiner Loops

We study a variety of loops, RIF, which arise naturally from considering inner mapping groups, and a somewhat larger variety, ARIF. All Steiner and Moufang loops are RIF, and all flexible C-loops are ARIF. All ARIF loops are diassociative. Received June 6, 2001; accepted in final form November 14, 2002.     

Sylow Theory for Quasigroups II: Sectional Action

The first paper in this series initiated a study of Sylow theory for quasigroups and Latin squares based on orbits of the left multiplication group. The current paper is based on so‐called pseudo‐orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo‐orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well behaved in the newly identified class of conflatable quasigroups, which provides a unified treatment of Moufang, Bol, and conjugacy closure properties. Relationships between sectional and Lagrangean properties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo‐orbits. An upper bound is given on the size of a pseudo‐orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invariants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15.     

When is the commutant of a Bol loop a subloop?

A left Bol loop is a loop satisfying . The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order , odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to , the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop such that is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order with a non-subloop commutant.

     

Independent axiom systems for nearlattices

A nearlattice is a join semilattice such that every principal filter is a lattice with respect to the induced order. Hickman and later Chajda et al independently showed that nearlattices can be treated as varieties of algebras with a ternary operation satisfying certain axioms. Our main result is that the variety of nearlattices is 2-based, and we exhibit an explicit system of two independent identities. We also show that the original axiom systems of Hickman as well as that of Chajda et al are dependent.