Inverse Semigroup - E-unitary Inverse Semigroups | <i>E< I>-unitary Inverse Semigroups

E-unitary Inverse Semigroups

One class of inverse semigroups which has been studied extensively over the years is the class of E-unitary inverse semigroups: an inverse semigroup S (with semilattice E of idempotents) is E-unitary if, for all e in E and all s in S,

Equivalently,

One further characterisation of an E-unitary inverse semigroup S is the following: if e is in E and e ≤ s, for some s in S, then s is in E.

Theorem. Let S be an inverse semigroup with semilattice E of idempotents, and minimum group congruence σ. Then the following are equivalent:

S is E-unitary;
σ is idempotent pure;
= σ,

where is the compatibility relation on S, defined by

are idempotent.

McAlister's Covering Theorem. Every inverse semigroup S has a E-unitary cover; that is there exists an idempotent separating surjective homomorphism from some E-unitary semigroup T onto S.

Central to the study of E-unitary inverse semigroups is the following construction. Let be a partially ordered set, with ordering ≤, and let be a subset of with the properties that

is a lower semilattice, that is, every pair of elements A, B in has a greatest lower bound A B in (with respect to ≤);
is an order ideal of, that is, for A, B in, if A is in and B ≤ A, then B is in .

Now let G be a group which acts on (on the left), such that

for all g in G and all A, B in, gA = gB if, and only if, A = B;
for each g in G and each B in, there exists an A in such that gA = B;
for all A, B in, A ≤ B if, and only if, gA ≤ gB;
for all g, h in G and all A in, g(hA) = (gh)A.

The triple is also assumed to have the following properties:

for every X in, there exists a g in G and an A in such that gA = X;
for all g in G, g and have nonempty intersection.

Such a triple is called a McAlister triple. A McAlister triple is used to define the following:

together with multiplication

Then is an inverse semigroup under this multiplication, with (A,g)−1 = (g−1A, g−1). One of the main results in the study of E-unitary inverse semigroups is McAlister's P-Theorem:

McAlister's P-Theorem. Let be a McAlister triple. Then $P(G, \mathcal{X}, \mathcal{Y})$ is an E-unitary inverse semigroup. Conversely, every E-unitary inverse semigroup is isomorphic to one of this type.