Discrimination of organic acids using a three molecule array based upon cruciform fluorophores

A small array was obtained from three reactive cruciform fluorophores in six different solvents. The array discerned 10 different aromatic carboxylic acids by protonation-induced fluorescence shifts, which were recorded by digital photography. This simple array can discern acids that have closely spaced pK(a) values.     

Full does not imply strong, does it?

We give a duality for the variety of bounded distributive lattices that is not full (and therefore not strong) although it is full but not strong at the finite level. While this does not give a complete solution to the “Full vs Strong” Problem, which dates back to the beginnings of natural duality theory in 1980, it does solve it at the finite level. One consequence of this result is that although there is a Duality Compactness Theorem, which says that if an alter ego of finite type yields a duality at the finite level then it yields a duality, there cannot be a corresponding Full Duality Compactness Theorem. Received October 1, 2002; accepted in final form November 10, 2004.     

Not every full duality is strong!

Is every full duality strong? We show that the answer is ‘no’, thereby answering a question that goes back to the earliest foundations of the theory of natural dualities. Received July 3, 2006; accepted in final form September 8, 2006.     

Full but not strong dualities at the finite level: Extending the realm

The realm of natural dualities that are known to be full but not strong at the finite level is a very small one, consisting of a single example. This example, based on the three-element bounded distributive lattice, was presented by Davey, Haviar and Willard [8]. In this paper, we extend this realm to the class of all natural dualities based on a finite non-boolean bounded distributive lattice. Received June 15, 2005; accepted in final form November 26, 2005.     

Binary homomorphisms and natural dualities

We investigate ways in which certain binary homomorphisms of a finite algebra can guarantee its dualisability. Of particular interest are those binary homomorphisms which are lattice, flat-semilattice or group operations. We prove that a finite algebra which has a pair of lattice operations amongst its binary homomorphisms is dualisable. As an application of this result, we find that every finite unary algebra can be embedded into a dualisable algebra. We develop some general tools which we use to prove the dualisability of a large number of unary algebras. For example, we show that the endomorphisms of a finite cyclic group are the operations of a dualisable unary algebra.     

From the subalgebras of the square to the discriminator

Presented by A. F. Pixley.     

When is a natural duality ‘good’?

Building on the most current work in the theory of natural dualities, we continue the study of strong dualities for the quasi-variety generated by a finite algebra. We investigate ten different versions of what we would like to mean by a good duality. Each version concerns, among other things, a specific restriction on the type of the structures in the dual category which insures that the dual structures will in a useful sense be simple. Through each investigation we seek a theorem characterizing, in terms of finitely verifiable conditions, those finite algebras generating a quasi-variety which admits a strong duality meeting the given restrictions. Our study includes a careful treatment of coproducts, logarithmic dualities and strong dualities by various unary structures.Dedicated to the memory of Alan DayPresented by J. Sichler.Research supported by a 1992 ARC Grant (Davey).     

A non-dualisable entropic algebra

We give an example of a five-element unary algebra which is entropic but does not admit a natural duality. Received September 26, 2000; accepted in final form April 4, 2001.