Equationally complete discriminator varieties of groupoids

Abstract:

J. Kalicki proved that there are continuum many equationally complete varieties of groupoids. In this note we give a constructive proof of this by defining a countable partial groupoid which has continuum many completions such that each completion generates an equationally complete variety, and no two distinct completions generate the same variety. Moreover, the variety generated by all the completions is a discriminator variety, and every nontrivial groupoid in this variety is cancellative but not a quasigroup; this answers a question of R. Padmanabhan. A. D. Bol’bot proved a similar result for loops, but his computations are more difficult since his varieties are not discriminator varieties.