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A commutative semigroup S
has property (α) if (1) S is topologically a two-cell, (2) S has no zero divisors, and
(3) the boundary of S is the union of two unit intervals with the usual multiplication.
A characterization of semigroups having property (α) will be given. Let (I,⋅)
denote the closed unit interval with the usual multiplication. Let M be a
closed ideal of (I,⋅) × (I,⋅) such that M contains (I ×{0}) ∪ ({0}× I), and
M ∩ (I ×{1}) = {(0,1)} or M ∩ ({1}× I) = {(1,0)}. For each a,b ∈ (0,1) define a
relation R(a,b;M) on (I,⋅) × (I,⋅) by (x,y) ∈ R(a,b;M) if (1) x = y or (2)
x,y ∈ (I ×{0}) ∪ ({0}×I), or (3) there exists an s ∈ (0,∞) such that x and y are in
the same component of M ∩{(ast,bs−st) : 0 ≦ t ≦ 1}. LEMMA. The relation
R(a,b;M) is a closed congruence. THEOREM. A semigroup S has property (α) if
and only if there exists a,b,M such that (I,⋅) × (I,⋅)∕R(a,b;M) is iseomorphic to
lS.
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