Ordered sets which support inverse semigroups: three examples (Q800391)

A partially ordered set (A,\(\leq)\) is said to support an inverse semigroup if there exists an inverse semigroup (A,\(\cdot)\) for which \(\leq\) is the natural partial order. \textit{V. V. Vagner} [Mat. Sb., Nov. Ser. 32(74), 545-632 (1953; Zbl 0053.206)] gave a result that if a and b are upper bounded elements of S, then a and b have a greatest lower bound. The authors give (1) an example of a partially ordered set which satisfies Vagner's condition but does not support any inverse semigroup, (2) an example of a partially ordered set which supports an inverse semigroup but does not support any inverse semigroup which is a semilattice of groups, and (3) an example of a partially ordered set which supports an inverse semigroup which is a semilattice of groups but does not support any commutative inverse semigroup.