Lathes with a Simplex badge were both a native American product, from the "Cleveland" Simplex Machine Tool Company of, Hamilton Ohio (established in the late 1800s and who later absorbed the Kern Machine Tool Company of East Hamilton) and by Mondiale, a Belgium Company whose range during the 1950s included machines with a Simplex logo on their headstock or stand.
Although the only available pictures of the American lathes are shown below, it known that several examples have survived until recent times and, if photographs of these (or catalog illustrations) can be supplied by readers, they will be added to this section. Please contact the author if you are able to help.
It is know that the English Importers and dealers Selson bought numbers of geared-head Simplex machines during the 1930s, some carrying simple screwed-on badges others (obviously part of a bigger batch), with the Selson name cast into the bed..

An American Simplex, made during the 1930s, badged as a "Selson" and as sold in the United Kingdom. Note the lever, immediately to the right of the screwcutting gearbox, a Simplex "trade-mark" feature also found on their earlier models

An other example of a Selson lathe  manufactured by Simplex. This model was  branded, in the United Kingdom, as the "Regent"