Holoidal compactifications of uniquely divisible semigroups



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A semigroup T is uniquely divisible if, for each x [element of] T and each positive integer n, there exists a unique element y [element of] T such that y[raised n] = x. A holoid is a semigroup with degenerate subgroups. If T is a compact uniquely divisible holoid, the thread generated by a non-idempotent element x is the closure of {x[raised r]|r a positive rational number}, and any such entity is topologically isomorphic to the real unit interval [0,1] under multiplication. Suppose that T is a uniquely divisible, compact, connected holoid such that E = {0,1}. Let S = T{0}, then S is a uniquely divisible holoid. This paper is concerned generally with holoidal compactifications of S. First two maximal holoidal compactifications of S are constructed; the maximal uniquely divisible holoidal compactification, (h,H(S)), of S and the maximal uniquely divisible holoidal kernel compactification, (q,Q(S)), of S. The latter compactification has the property that Q(S)q(S) is the minimal ideal of Q(S) and, if K is a compact, uniquely divisible holoid and g:S------>K is an iseomprphism into ,K such that Kg(S) is the minimal ideal of K, then there exists a continuous homomorphism g[lowered 1]:Q(S)------>K such that g[lowered 1]q = g. Chapter IV is concerned with the case in which S is cancellative and satisfies xS [subset] Sx for all x [element of] S. One of the main theorems of this chapter states a necessary and sufficient algebraic condition on S for distinct threads of T to have distinct zeros in H(S). This condition is: if A is a subset of S such that Ax = xA, x [not equal] 1, then A is contained in the thread generated by x. Finally, if T contains a normal thread N, then application of a standard congruence on S yields a nontrivial ideal compactification of S in some cases. This compactification is constructed by a new method involving adjoining a quotient of S to S and the subsequent replacement of the zero. Requirements on T for this to be the case involve the separation of N{1} and a closed uniquely divisible semigroup containing a certain set of coset representatives.