Anti-uniform semilattices
John M Howie, Boris M. Schein · Bulletin of the Australian Mathematical Society · 1969
An inverse semigroup which is a union of groups is called Cliffordian. A semilattice E is called universally Cliffordian if every inverse semigroup having E as semilattice of idempotents is Cliffordian. It is shown that E is universally Cliffordian if and only if it is anti-uniform, that is, if and only if no two distinct principal ideals of E are isomorphic. A semilattice E satisfying the minimum condition is anti-uniform if and only if it is a well-ordered chain. Examples are given of anti-uniform semilattices of more complicated types.