Vol. 34, No. 3, 1970

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ISSN: 0030-8730
Special semigroups on the two-cell

Esmond Ernest Devun

Vol. 34 (1970), No. 3, 639–645
Abstract

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}) ({0I), and M (I ×{1}) = {(0,1)} or M ({1I) = {(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}) ({0I), or (3) there exists an s (0,) such that x and y are in the same component of M ∩{(ast,bsst) : 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.

Mathematical Subject Classification
Primary: 22.05
Secondary: 54.00
Milestones
Received: 17 February 1969
Published: 1 September 1970
Authors
Esmond Ernest Devun