Example of a nonacyclic continuum semigroup $S$ with zero and $S=ESE$

Anne L. Hudson
1963 Proceedings of the American Mathematical Society  
Throughout this discussion S will denote a compact connected topological semigroup and E will denote the set of idempotents of S. The problem to be considered concerns a question posed by Professor A. D. Wallace. In [l], Wallace proves that if 5 has a left unit, if I is a closed ideal of S, and if F = Q or if F is a closed left ideal of S, then HniS)^HniiyJL) for all integers n, where HniA) denotes the wth Alexander-Cech cohomology group of A with coefficients in an arbitrary but fixed group G.
more » ... but fixed group G. If 5 is assumed to have both a left zero and a left unit, then it follows that each closed left ideal L, of 5 is acyclic; that is, HpiL)=0 for all p^l. A dual statement holds for closed right ideals if S has a right unit and right zero. A generalization of the case in which 5 has a left, right, or two-sided unit, is to require that S=ES, S=SE, or S = ESE, respectively, and Wallace has asked : "If S has a zero, are closed right or left ideals of 5 necessarily acyclic in the more general situation?" [3]. A negative answer to this question is given here by way of examples, and a theorem is proved giving a necessary and sufficient condition for closed right ideals of 5 to be acyclic, assuming S = ESE and 5 has a zero. Following the proof of this theorem is an example of a semigroup not satisfying this condition. The above-mentioned example shows that even though 5 is acyclic, it is not necessarily true that all closed right ideals of 5 are acyclic. Thus the question remains as to whether S is acyclic if S = ESE and S has a zero [3]. Wallace proves in [2] that for such a semigroup S, H1iS) = 0, however, an example is given here of a semigroup 5 with zero, S = ESE and H2iS)=G for all groups G, showing that this question also has a negative answer. Two further examples are included in this paper which show what can occur if one only assumes that S = SE, or S = ES. One example is of a semigroup 5 with zero, S=SE and HliS)=G for all groups G and the other is an example of a semigroup 5 with zero and left unit and 5 contains a closed right ideal R with IP(R)^G. Definition. Let F be a semigroup, aE T and Rio) the closed right ideal of T generated by ex. Then a is said to be right codependent on Received by the editors May 2, 1962.
doi:10.1090/s0002-9939-1963-0151940-x fatcat:yq7dmnn7bfex7lc7hppgzmyzxm