Semigroups with quasizeroes

Let S be a semigroup. An element a ∈ S is said to be a left quasi-zero if x ⋂ ≠ ∅ for all x ∈ S, where denotes the cyclic sub-semigroup of S generated by a. In a recent study [6] of semigroups with a maximum right congruence, such elements proved to be useful in providing characterizations of these semigroups. Left quasizeroes have appeared in the literature under different names in a variety of situations. In the context of semigroup radicals, left quasi-zeroes are called right quasi-regular elements, where an element is defined to be right quasi-regular if it is not a left identity for any right congruence other than the universal congruence (see [4], [5], [2], [7], and [8]).