Special semigroups on the two-cell
E. E. DeVun · Pacific Journal of Mathematics · 1970
A commutative semigroup £ 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 (a) will be given.Let (7, •) denote the closed unit interval with the usual multiplication.Let I be a closed ideal of (I, •) X (7, •) such that M contains (7 X {0}) u ({0} X 7), and M n (7 x {1}) = {(0,1)} or M n ({1} x 7) = {(1, 0)}.For each a, be (0,1) define a relation R(a, b; M) on (7, •) x (7, •) by (x, y) 6 R(a, b; M) if (1) x = y or (2) x,ye(Ix {0}) u ({0} X 7), or (3) there exists an s e (0, oo) such that x and y are in the same component of M n {(α sί , b s ~st ): 0 ^ t ^ 1}.LEMMA.The relation R(a, b; M) is a closed congruence.THEOREM.A semigroup S has property (a) if and only if there exists a, b, M such that (7, •) x (7, )IR(a, 6; M) is iseomorphic to S.