×

Symmetric inverse semigroups. (English) Zbl 0857.20047

Mathematical Surveys and Monographs. 46. Providence, RI: American Mathematical Society (AMS). xviii, 166 p. (1996).
The symmetric inverse semigroup \(I_X\) on a set \(X\) is the semigroup of all partial one-to-one mappings on \(X\) under composition. It plays the role in inverse semigroup theory corresponding to that of the symmetric group in group theory: every inverse semigroup \(S\) may be embedded in \(I_S\) (a semigroup \(S\) is inverse is each \(a\in S\) has a unique inverse, \(a^{-1}\), in the sense that \(a=aa^{-1}a\) and \(a^{-1}=a^{-1}aa^{-1}\)).
The book is dedicated to the study of the finite symmetric inverse semigroup \(C_n\) on a base set of size \(n\) whose members are here called charts. Connections with abstract semigroup theory are confined to the background and are only called upon as they arise. There is an appendix which briefly recounts some relevant notions from group and semigroup theory, general algebra and categories.
The author places great store by his introduction of ‘path notation’ which is a generalization of cycle notation for permutations: any chart \(\alpha\) can be represented uniquely as a disjoint union of cycles and paths, paths being maps of the form \((i_1,i_2,\dots,i_kq]\), where \(q\not\in\text{dom }\alpha\). This necessitates writing 3 symbols for each point \(j\) not in the domain nor range of \(\alpha\): \((j]\). In practice though the notation seems a useful enough tool for formulation and calculation and is used in the opening two chapters to illustrate counterparts of well-known facts about the symmetric group: for example, two charts are conjugate if and only if they share the same path structure.
This monograph is not a textbook in that each of the 13 chapters does not close with a series of exercises but rather with a comments section generally giving a brief overview of the historical development of the topic in question.
The first pair of chapters is introductory while 3 through 5 are concerned with commuting charts and more particularly, centralizers of charts.
In Chapter 6 the notion of an even chart is introduced as one which is a product of an even number of transpositionals: charts which are either transpositions \((ij)\), or semitranspositions, \((ij]\). The even charts together comprise \(A^c_n\), the alternating semigroup, which is one of the so-called \(S_n\)-normal subsemigroups of \(C_n\) (permutationally self-conjugate subsemigroups). A full classification of the \(S_n\)-normal subsemigroups of \(C_n\) is given in Chapter 7 and the role they play in finding all the homomorphic images of \(C_n\) is determined in Chapter 8.
Chapters 9 and 10 are devoted to presenting \(C_n\) and \(A^c_n\) respectively by means of generators and relations and a general approach to the presentation problem for a semigroup consisting of all restrictions of a group of permutations with a given presentation is thereby developed.
The final three chapters depart from the exclusive study of \(C_n\) and instead look at \(PT_n\), the semigroup of all partial (not necessarily injective) mappings on a set. A path notation is developed in this wider context and various applications given including a reformulation of the graph reconstruction conjecture in terms of an extension problem of one Brandt semigroup by another.
Overall, this is an enthusiastic account, full of detail, worked examples, and pictures. Researchers in transformation semigroups will find it an accessible and useful book to dip into for facts and for ideas.
There is an index and bibliography of over 120 references.

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
20M18 Inverse semigroups
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20B35 Subgroups of symmetric groups
PDFBibTeX XMLCite