The smallest regular semigroups with all four skew pairs of idempotents. (Q706046)

An ordered pair of idempotents \((e,f)\) of a regular semigroup is called a skew pair if \(ef\) is not idempotent yet \(fe\) is. There are four distinct types of skew pairs known respectively as strong, left regular, right regular and discrete. This paper finds the smallest regular semigroups that feature exactly one of these four types. It transpires that, to within isomorphism and dual isomorphism, there are six such semigroups and each can be realised as particular quotients of certain regular Rees semigroups.