On the variety generated by all semigroups of order three

Denote by Sn the variety generated by all semigroups of order n. Marcel Jackson proved that the variety Sn contains uncountably many subvarieties if n⩾4, and it follows from existing results that the variety S2 contains precisely 32 subvarieties. However, the number of subvarieties of the variety S3 has been unknown. The main aim of the present article is to address this problem. It is shown that all subvarieties of the variety S3 are finitely based. Consequently, the variety S3 contains countably infinitely many subvarieties.