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- W2155691985 abstract "Let S be a compact semigroup and f a continuous homomorphism of S onto the (compact) semigroup T . What can be said concerning the relations among S , f , and T ? It is to one special aspect of this problem which we shall address ourselves. In particular, our primary considerations will be directed toward the case in which T is a standard thread. A standard thread is a compact semigroup which is topologically an arc, one endpoint being an identity element, the other being a zero element. The structure of standard threads is rather completely determined e.g. see [20]. Among the standard threads there are three which have a rather special rôle. These are as follows: A unit thread is a standard thread with only two idempotents and no nilpotent element. A unit thread is isomorphic to the usual unit interval [14]. A nil thread again has only two idempotents but has a non-zero nilpotent element. A nil thread is isomorphic with the interval [½, 1], the multiplication being the maximum of ½ and the usual product — or, what is the same thing, the Rees quotient of the usual [0, 1] by the ideal [0,½ ]. Finally there is the idempotent thread , the multiplication being x o y = mm ( x, y ). These three standard threads can often be considered separately and, in this paper, we reserve the symbols I 1 I 2 and I 3 to denote the unit, nil and idempotent threads respectively. Also, throughout this paper, by a homomorphism we mean a continuous homomorphism." @default.
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- W2155691985 date "1967-08-01" @default.
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- W2155691985 title "Certain homomorphisms of a compact semigroup onto a thread" @default.
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- W2155691985 doi "https://doi.org/10.1017/s1446788700004158" @default.
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