Around 1750 the cafes of Paris were the place for serious chess. The most famous of these was the Cafe de la Regence. Francois-Andre Philidor, and Paul Morphy were frequent patrons, along with Jefferson and Napoleon, to name just a few. The cafe was so famous the contemporary style of chess set became known as French Regence or Regency sets its honor. The name is ahistorical. People were playing with sets that looked like this long before the cafe was hot and the design had a name. 20th century suggestions that anything before the cafe should be called 'directoroire' or some other name are mostly silly, in my opinion. It's the same design. We call it Regence. If you object because they didn't call it Regence before the cafe... well they didn't call it Directoire either. Before Jaques made a big deal of creating a new 'revolutionary' chess set design sets this is the way chess set naming worked... mostly ahistorically. You can call sets with bone ornaments 'Lyon' if you like. I like that (despite the fact that no one has any evidence those sets were made in Lyons, France,) but only if Lyon is a kind of Regence set because anyone can see the similarity. The popularity of this style of set lived on through the nineteenth and early twentieth century. Of course they were replaced by the Staunton set when chess sets became standardized in that design, but the transition was gradual and Regence sets were slower to fall out of fashion than other designs. This is why, of all of the popular pre-Staunton designs (St. George, Barleycorn, Selenus, etc....,) Regence sets are the most common and easy to come by today. Twentieth century boxwood Regence sets are all over eBay at any given day. Don't let anyone tell you they are rare or 19th century. The design has a simple elegance about it. The shape, feel, and balance make them excellent for play. Benjamin Franklin famously owned a wooden Regence set with plain turned knights. That particular design is often referred to as the 'Franklin Set' in his honor.
What are the Mexican sets doing here? Mexico was occupied by France under Emperor Maximilian from 1864 to 1867. This brief reign resulted in a lifelong mark on Mexican chess sets. French Lyon sets inspired Mexicans to assemble wood and bone and the shape of many Mexican sets resemble early Regence sets. I have included a few French influenced Mexican sets in this section. The rest of my Mexican sets are in the Spanish Pulpit section.
I have ten Regence sets and one orphaned bishop.