“Inverse semigroup”的意思、由来-开放百科全书

In group theory, an inverse semigroup (occasionally called an inversion semigroup^[1]) S is a semigroup in which every element x in S has a unique inverse y in S in the sense that x = xyx and y = yxy, i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.^[2]

(The convention followed in this article will be that of writing a function on the right of its argument, and

composing functions from left to right—a convention often observed in semigroup theory.)

Origins

Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner^[3] in the Soviet Union in 1952,^[4] and by Gordon Preston in Great Britain in 1954.^[5] Both authors arrived at inverse semigroups via the study of partial one-to-one transformations of a set: a partial transformation α of a set X is a function from A to

B, where A and B are subsets of X. Let α and β be partial transformations of a set X; α and β can be composed (from left to right) on the largest domain upon which it "makes sense" to compose them:

where α⁻¹ denotes the preimage under α. Partial transformations had already been studied in the context of pseudogroups.^[6] It was Wagner, however, who

was the first to observe that the composition of partial transformations is a special case of the composition of binary relations.^[7] He recognised also that the domain of composition of two partial transformations may be the empty set, so he introduced an empty transformation to take account of this. With the addition of this empty transformation, the composition of partial transformations of a set becomes an everywhere-defined associative binary operation. Under this composition, the collection

of all partial one-one transformations of a set X forms an inverse semigroup, called the symmetric inverse semigroup (or monoid) on X, with inverse the functional inverse defined from image to domain (equivalently, the converse relation).^[8] This is the "archetypal" inverse semigroup, in the same way that a symmetric group is the archetypal group. For example, just as every group can be embedded in a symmetric group, every inverse semigroup can be embedded in a symmetric inverse semigroup (see "Homomorphisms and representations of inverse semigroups" below).

The basics

The inverse of an element x of an inverse semigroup S is usually written x⁻¹. Inverses in an

inverse semigroup have many of the same properties as inverses in a group, for example,

x⁻¹x are not necessarily equal to the identity, but they are both idempotent.^[9] An inverse monoid S in which xx⁻¹ = 1 =

There are a number of equivalent characterisations of an inverse semigroup S:^[10]

Unless stated otherwise, E(S) will denote the semilattice of idempotents of an inverse semigroup S.

Examples of inverse semigroups

The natural partial order

An inverse semigroup S possesses a natural partial order relation ≤ (sometimes denoted by ω)

for some (in general, different) idempotent f in S. In fact, e can be taken to be

The natural partial order is compatible with both multiplication and inversion, that is,^[14]

In a group, this partial order simply reduces to equality, since the identity is the

only idempotent. In a symmetric inverse semigroup, the partial order reduces to restriction of mappings,

i.e., α ≤ β if, and only if, the domain of α is contained in the domain of β and

The natural partial order on an inverse semigroup interacts with Green's relations as follows: if s ≤

so, since the idempotents form a semilattice under the product operation, products on E(S) give least upper bounds with respect to

If E(S) is finite and forms a chain (i.e., E(S) is totally ordered by ≤), then

Homomorphisms and representations of inverse semigroups

A homomorphism (or morphism) of inverse semigroups is defined in exactly the same way as for any other

morphism of inverse semigroups could be augmented by including the condition (sθ)⁻¹ =

s⁻¹θ, however, there is no need to do so, since this property follows from the above

Theorem. The homomorphic image of an inverse semigroup is an inverse semigroup; the

One of the earliest results proved about inverse semigroups was the Wagner–Preston Theorem, which is an analogue

Wagner–Preston Theorem. If S is an inverse semigroup, then the function φ

Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup, and with image closed under the inverse operation on partial bijections. Conversely, any subsemigroup of the symmetric inverse semigroup closed under the inverse operation is an inverse semigroup. Hence a semigroup S is isomorphic to a subsemigroup of the symmetric inverse semigroup closed under inverses if and only if S is an inverse semigroup.

Congruences on inverse semigroups

Congruences are defined on inverse semigroups in exactly the same way as for any other semigroup: a

congruence ρ is an equivalence relation which is compatible with semigroup multiplication, i.e.,

It can be shown that σ is a congruence and, in fact, it is a group congruence, meaning that the factor semigroup S/σ is a group. In the set of all group congruences on a semigroup S, the minimal element (for the partial order defined by inclusion of sets) need not be the smallest element. In the specific case in which S is an inverse semigroup σ is the smallest congruence on S such that S/σ is a group, that is, if τ is any

other congruence on S with S/τ a group, then σ is contained in τ. The congruence σ is

congruence can be used to give a characterisation of E-unitary inverse semigroups (see below).

E-unitary inverse semigroups

One class of inverse semigroups which has been studied extensively over the years is the class of E-unitary

inverse semigroups: an inverse semigroup S (with semilattice E of idempotents) is

One further characterisation of an E-unitary inverse semigroup S is the following: if e is in E and

Theorem. Let S be an inverse semigroup with semilattice E of idempotents, and minimum group

Central to the study of E-unitary inverse semigroups is the following construction.^[29] Let

be a partially ordered set, with ordering ≤, and let

be a subset of

with the properties that

McAlister's P-Theorem. Let

be a McAlister triple. Then

is an E-unitary inverse semigroup. Conversely, every E-unitary inverse

F-inverse semigroups

An inverse semigroup is said to be F-inverse if every element has a unique maximal element above it in the natural partial order, i.e. every σ-class has a maximal element. Every F-inverse semigroup is an E-unitary monoid. McAlister's covering theorem has been refined by M.V. Lawson to:

McAlister's P-theorem has been used to characterize F-inverse semigroups as well. A McAlister triple

is an F-inverse semigroup if and only if

is a principal ideal of

and

is a semilattice.

Free inverse semigroups

A construction similar to a free group is possible for inverse semigroups. A presentation of the free inverse semigroup on a set X may be obtained by considering the free semigroup with involution, where involution is the taking of the inverse, and then taking the quotient by the Vagner congruence

The word problem for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to W. D. Munn who showed that elements of the free inverse semigroup can be naturally regarded as trees, known as Munn trees. Multiplication in the free inverse semigroup has a correspondent on Munn trees, which essentially consists of overlapping common portions of the trees. (see Lawson 1998 for further details)

Connections with category theory

The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There is another way of composing partial transformations, which is more restrictive than that used above: two partial transformations α and β are composed if, and only if, the image of α is equal to the domain of β; otherwise, the composition αβ is undefined. Under this alternative composition, the collection

of all partial one-one transformations of a set forms not an inverse semigroup but an inductive groupoid, in the sense of category theory. This close connection between inverse semigroups and inductive groupoids is embodied in the Ehresmann–Schein–Nambooripad Theorem, which states that an inductive groupoid can always be

constructed from an inverse semigroup, and conversely.^[32] More precisely, an inverse semigroup is precisely a groupoid in the category of posets which is an étale groupoid with respect to its (dual) Alexandrov topology and whose poset of objects is a meet-semilattice.

Generalisations of inverse semigroups

As noted above, an inverse semigroup S can be defined by the conditions (1) S is a regular semigroup,

and (2) the idempotents in S commute; this has led to two distinct classes of generalisations of

an inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa.

Inverse category

This notion of inverse also readily generalizes to categories. An inverse category is simply a category in which every morphism f:X→Y has a generalized inverse g:Y→X such that fgf = f and gfg = g. An inverse category is selfdual. The category of sets and partial bijections is the prime example.^[36]

Inverse categories have found various applications in theoretical computer science.^[37]

See also

Notes

1. ^{{cite book |first=Eric W. |last=Weisstein|title=CRC Concise Encyclopedia of Mathematics |edition=2nd |url=https://books.google.com/books?id=aFDWuZZslUUC&pg=PA1528|year=2002|publisher=CRC Press|isbn=978-1-4200-3522-3|page=1528}}
2. ^{{harvnb|Lawson|1998}}
3. ^Since his father was German, Wagner preferred the German transliteration of his name (with a "W", rather than a "V") from Cyrillic — see {{harvnb|Schein|1981}}.
4. ^First a short announcement in {{harvnb|Wagner|1952}}, then a much more comprehensive exposition in {{harvnb|Wagner|1953}}.
5. ^{{harvnb|Preston|1954a}},b,c.
6. ^See, for example, {{harvnb|Gołab|1939}}.
7. ^{{harvnb|Schein|2002|p=152}}
8. ^{{harvnb|Howie|1995|p=149}}
9. ^{{harvnb|Howie|1995|loc=Proposition 5.1.2(1)}}
10. ^{{harvnb|Howie|1995|loc=Theorem 5.1.1}}
11. ^{{harvnb|Howie|1995|loc=Proposition 5.1.2(1)}}
12. ^{{harvnb|Wagner|1952}}
13. ^{{harvnb|Howie|1995|loc=Proposition 5.2.1}}
14. ^{{harvnb|Howie|1995|pp=152–3}}
15. ^{{harvnb|Howie|1995|p=153}}
16. ^{{harvnb|Lawson|1998|loc=Proposition 3.2.3}}
17. ^{{harvnb|Clifford|Preston|1967|loc=Theorem 7.5}}
18. ^{{Cite journal|last=Gonçalves|first=D|last2=Sobottka|first2=M|last3=Starling|first3=C|date=2017|title=Inverse semigroup shifts over countable alphabets|url=|journal=Semigroup Forum|volume=96|issue=2|pages=203–240|doi=10.1007/s00233-017-9858-5|postscript=Corollary 4.9|via=|arxiv=1510.04117}}
19. ^{{harvnb|Clifford|Preston|1967|loc=Theorem 7.36}}
20. ^{{harvnb|Howie|1995|loc=Theorem 5.1.7}} Originally, {{harvnb|Wagner|1952}} and, independently, Preston 1954c.
21. ^{{harvnb|Howie|1995|p=22}}
22. ^{{harvnb|Lawson|1998|p=62}}
23. ^{{harvnb|Lawson|1998|loc=Theorem 2.4.1}}
24. ^{{harvnb|Lawson|1998|p=65}}
25. ^{{harvnb|Howie|1995|p=192}}
26. ^{{harvnb|Lawson|1998|loc=Proposition 2.4.3}}
27. ^{{harvnb|Lawson|1998|loc=Theorem 2.4.6}}
28. ^{{cite book |first=P. A. |last=Grillet |title=Semigroups: An Introduction to the Structure Theory |url=https://books.google.com/books?id=yM544W1N2UUC&pg=PA248 |date=1995 |publisher=CRC Press |isbn=978-0-8247-9662-4 |page=248}}
29. ^{{harvnb|Howie|1995|pp=193–4}}
30. ^{{harvnb|Howie|1995|loc=Theorem 5.9.2}}. Originally, {{harvnb|McAlister|1974a}},b.
31. ^¹{{harvnb|Lawson|1998|p=230}}
32. ^{{harvnb|Lawson|1998|loc=4.1.8}}
33. ^{{harvnb|Howie|1995|loc=Section 2.4 & Chapter 6}}
34. ^{{harvnb|Howie|1995|p=222}}
35. ^{{harvnb|Fountain 1979}}, {{harvnb|Gould}}
36. ^{{cite book |first=Marco |last=Grandis |title=Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups |url=https://books.google.com/books?id=TWqhelao4KsC&pg=PA55 |year=2012 |publisher=World Scientific |isbn=978-981-4407-06-9 |page=55}}
37. ^{{cite book |editor-first=Simon |editor-last=Gay and |editor2-first=Ian |editor2-last=Mackie |title=Semantic Techniques in Quantum Computation |chapterurl=https://books.google.com/books?id=__LnWv6NYM0C&pg=PA369 |year=2010 |publisher=Cambridge University Press |isbn=978-0-521-51374-6 |page=369 |first=Peter |last=Hines |first2=Samuel L. |last2=Braunstein |chapter=The Structure of Partial Isometries}}