New variations on the theme

Variations on a theme.

A ring satisfying x = x is necessarily commutative. We consider a variety of weaker forms of this condition and show that many but not all of them imply commutativity. We also present a variety of elementary proofs of the fact that x = x implies commutativity.

Variations on a Theme

Variations on a Montagovian theme

What are the objects of knowledge, belief, probability, apriority or analyticity? For at least some of these properties, it seems plausible that the objects are sentences, or sentence-like entities. However, results from mathematical logic indicate that sentential properties are subject to severe formal limitations. After surveying these results, I argue that they are more problematic than ofte...

The Klein Bottle: Variations on a Theme

It is a well-known fact that K is a genus 2 nonorientable closed surface with Euler characteristic χ = 0, which is topologically equivalent to a couple of Möbius bands glued together along the border, and that it is nonembeddable in R. It is possible, however, to immerse it in R, that is, to map it into R obtaining an image with no singular points. To give an immersion of K in R, it suffices to...

Theme and Variations on the Concatenation Product

The concatenation product is one of the most important operations on regular languages. Its study requires sophisticated tools from algebra, finite model theory and profinite topology. This paper surveys research advances on this topic over the last fifty years. The concatenation product plays a key role in two of the most important results of automata theory: Kleene’s theorem on regular langua...