Lumon began as a salve company and then moved into medical devices and technology. They still make their own soap, a heavy, viscous goo of a bright green color reminiscent of Palmolive with a gelatin thickener. That is the good soap that is refilled each morning by the Senior Refiner. There is also the bad soap which is used by security personnel on Severed employees who have misbehaved. You don't want them to use the bad soap.
Some think that the topical application of the soap enhances the compliance of innies, but there is some prescription of compulsion to use the soap regularly, one pump, ten times a day. As Irving points out, the soap dispensers are not labeled, though everything else in the innie universe is. Perhaps that means that the label “soap” is disingenuous?
Imogene's is a soap store in the town of Kier, named for the wife of Kier Eagan. Imogene Eagan was a swab girl (who cleans the vats) in the ether factory where she met Kier. It seems clear from the emphasis of purity of soul and purging oneself of the Tempers (burdensome emotions, see Tempers) that Kier reveres the idea of soap. Does this reverence stem from his love for Imogen, or did he love her because she represented pureness?
Lumon also appears to have older logos that have evolved into the simple drop logo currently associated with the company.