By Tammo tom Dieck

ISBN-10: 3540505288

ISBN-13: 9783540505280

This e-book is a jewel– it explains very important, priceless and deep issues in Algebraic Topology that you simply won`t locate in other places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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Extra resources for Algebraic Topology and Transformation Groups

Sample text

Clearly, e = e A e for each element e e P. Thus (P, A) is an inverse semigroup in which every element is idempotent. The converse is immediate by Proposition 8. ■ The properties of the natural partial order lead to an alternative charac­ terisation of groups. Proposition 10 Let S be an inverse semigroup. Then the natural partial order is the equality relation if, and only if, S is a group. Proof Suppose that the natural partial order is the equality relation. If e and / are two idempotents then ef < e, f.

Infinite distributivity is a straightforward consequence of the description of the join. The function t(s) = [s] is well-defined because [s] is a permissible subset by Lemma 14. It is easy to check that it is an injective homomorphism. If A € C(S) then A = |J{[a]: a 6 A}. Thus every element of C(S) is a join of a non-empty compatible subset of i{S). ■ The universal property of the embedding t: S -> C{S) is described below. Theorem 24 If 6: S —> T is any homomorphism to a complete, infinitely distributive inverse semigroup, then there is a unique join-preserving homo­ morphism 6*: C(S) -► T such that 6*L = 6 Proof Clearly any homomorphism preserves the compatibility relation.

See [382] for Wagner's connection with this book, and for Ehresmann's connection see page 337 of Volume II-1 of [71]. Although Preston's paper does not mention it explicitly, Preston was also motivated by the theory of pseudogroups, as he told me in a conversa­ tion at the Hobart Semigroup Conference in 1994. Preston's supervisor was J. H. C. Whitehead. An account of Wagner's work together with a complete bibliography may be found in [382]. 2 The notion of structures compatible with a pseudogroup of transformations may be found in a paper of Ehresmann dated 1947 (this is work number 20 in the list of papers to be found in [71]).

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