The universal compact subunithetic semigroup

Abstract

Compact subunithetic semigroups have been studied in [3] and [4], and related results can be found in [5], [6], and [7]. The structure of compact subunithetic semigroups is completely determined in this paper by exhibiting a universal compact subunithetic semigroup in the continuous homomorphism sense and a universal compact unithetic semigroup in the embedding sense. Generalizations of some of the results of [4] to include nonabelian semigroups are obtained. If S is a compact [uniquely] divisible semigroup and xES, then there exists a [unique] minimal compact divisible subsemigroup S(x) of S which contains x. Moreover, each such S(x) is subunithetic [unithetic]. Thus the study of the structure of compact subunithetic semigroups is essential to the study of compact divisible semigroups. NOTATION. The following notation will be used throughout this paper: 1. N =set of all positive integers; 2. Q=discrete additive semigroup of positive rationals; 3. I= [0, 1] with usual multiplication and topology; 4. =a-adic solenoid with a = (2, 3, * * ) [2, p. 114]; 5. 2 = universal compact solenoidal group [2, 25.19]; 6. 'J =universal compact solenoidal semigroup [6, II]. A semigroup S is said to be [uniquely] divisible if for each y E S and each nEN, there exists an [unique] element x ES such that y = x8. A topological semigroup T is said to be subunithetic if T contains a dense homomorphic image of Q (Note that T is divisible and abelian). A subunithetic semigroup T is said to be unithetic if T is uniquely divisible. If T is a unithetic semigroup and o: Q-->T is a homomorphism such that cr(Q) is dense in T, then the element x = o-(1) is called a unithetic generator of T. (Note that the rational powers of x are dense in T.) If S is a uniquely divisible topological semigroup and xES, then the subsemigroup [x] = {xr: r&Q}I (closure in S) is the unithetic subsemigroup of S generated by x. Note that S is unithetic if and only if S= [x] for some x cS.