Let S be an ordered inverse
semigroup, that is, an inverse semigroup with a simple order < which satisfies the
condition:
Let E be the subsemigroup of S constituted by all the idempotents of S. By a result
of Munn, Γ = S∕σ is an ordered group, where σ is the congruence relation such that
xσy if and only if ex = ey for some e ∈ E. An ordered inverse semigroup S is called
proper if the σclass I which is the identity element of Γ contains only idempotents of
S.
In a proper ordered inverse semigroup S, let Γ(e)(e ∈ E) be the set of those
members of Γ which intersect nontrivially with R_{e}. Each element of S can
be represented in the form (α,e), where e ∈ E and α ∈ Γ(e). We define
e^{α} = a^{−1}a ∈ E, where a = (α,e). Then Γ(e) and e^{α} satisfy the following six
conditions:
(i) ⋃
_{e∈E}Γ(e) = Γ;
(ii) I ∈ Γ(e) and e^{I} = e;
(iii) if f ≦ e in the semilattice with respect to the natural ordering of the
commutative idempotent semigroup E and α ∈ Γ(e), then α ∈ Γ(f) and f^{α} ≦ e^{α} in
the semilattice E;
(iv) if α ∈ Γ(e) and β ∈ Γ(e^{α}), then αβ ∈ Γ(e) and e^{αβ} = (e^{α})^{β};
(v) if α ∈ Γ(e), then α^{−1} ∈ Γ(e^{α});
(vi) if α ∈ Γ(e) ∩ Γ(f) and e ≦ f, then e^{α} ≦ f^{α}.
Also the product and the order in S determined by
Next we prove conversely a theorem asserting that, for an ordered commutative
idempotent semigroup E and an ordered group Γ, if Γ(e) and e^{α} satisfy the six
conditions above, then the set {(α,e);e ∈ E,α ∈ Γ(e)} is a proper ordered inverse
semigroup with respect to the product and the order mentioned above. Besides this,
we present other characterizations of special cases.
