Weakly strict regular semigroups. (Q2491172)

The strict regular semigroups are those that satisfy `\(\mathcal D\)-majorization': no idempotent is above distinct, \(\mathcal D\)-related idempotents. The author calls a regular semigroup `weakly strict' if whenever an idempotent is above distinct \(\mathcal D\)-related idempotents \(f\) and \(g\), either \(L_f\cap R_g\) or \(L_g\cap R_f\) contains an idempotent. (In contrast with the class \(\mathcal{RS}\) of strict regular semigroups, the class \(\mathcal{WS}\) of weakly strict regular semigroups is not an e-variety.) Not only does \(\mathcal{WS}\) contain \(\mathcal{RS}\), it contains the e-variety \(L\mathcal{CR}\) of locally completely regular semigroups. Noting that the class \(\mathcal{CR}\) of completely regular semigroups is the class of regular semigroups with the property that every e-variety not contained in \(\mathcal{CR}\) contains the 5-element combinatorial Brandt semigroup \(B_2\), it is proved that \(\mathcal{WS}\) is determined by the corresponding property with respect to the semigroup \(B_2^1\). In the remainder of the paper, the author proves theorems of this `exclusionary' type for a range of well known e-varieties of regular semigroups, along with appropriate e-identities for each. -- The relationships among the classes of semigroups considered in the paper are exhibited in a diagram.