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Given any discrete semigroup (S, +), there is an extension of the operation to PS making (PiS, +) a left topological semigroup. The aim of this paper is, among other things, to prove that there exist strictly increasing chains of principle left ideals and of principal closed ideals in (fiZ, +) . It is a question posed to M. E. Rudin several years ago by some analysts as to whether every point of 8iZ\Z = Z* is a member of some maximal orbit closure of the shift function. Here 8iZ is the Cech-Stone compactification of the group Z of integers and the "shift function" a refers both to the function from Z to Z defined by v(n) = n + 1 and to its continuous extension from fiZ to fiZ. The orbit closure of a point p of Z* is cl{an(p): n E Z} . A simpler question asks whether there can be any infinite strictly increasing chain of orbit closures. Of course a negative answer to the latter question implies an affirmative question to the former. It is well known that, given any discrete semigroup (S, +), there is an extension of the operation to 8S making (,6S, +) a left topological semigroup, with S contained in its topological center. That is to say, for each p E 8S the function Ap: /S -, fS defined by Ap(q) = p + q is continuous, while for s E S the function ps: f3S -* /S defined by ps(p) = p + s is continuous. (See, for example, [4].) In particular, in 8iZ we have for each n E Z that an and Pn are continuous functions agreeing on the dense set Z so that for all p E 8iZ and all n E Z, p + n = an(p). Consequently cl{an (p): n E Z} = cl{p + n: n E Z} = cl Ap [Z] = Ap [cl Z] = Ap [,8Z] = p + 8iZ. That is, the orbit closure of p is the principal right ideal generated by p. Consequently, the question we are addressing is: is there a strictly increasing chain ofprincipal right ideals in 8Z ? (Observe that if p E Z then p +,8Z = 8iZ, so asking for an increasing chain in 8iZ or in Z* amounts to the same thing.) Once the question is phrased in this fashion it immediately suggest two others: Received by the editors October 15, 1990. 1991 Mathematics Subject Classification. Primary 54D35, 22A30.

Increasing chains of ideals and orbit closures in 𝛽𝑍