|
|
| Line 1: |
Line 1: |
| In [[mathematics]], in [[semigroup theory]], an [[Involution (mathematics)|involution]] in a [[semigroup]] is a [[Transformation (geometry)|transformation]] of the semigroup which is its own [[Inverse function|inverse]] and which is an [[anti-automorphism]] of the semigroup. A semigroup in which an involution is defined is called a '''semigroup with involution''' or a '''*-semigroup.''' In the [[Matrix multiplication|multiplicative]] semigroup of [[Real number|real]] square [[Matrix (mathematics)|matrices]] of order ''n'', the [[Map (mathematics)|map]] which sends a matrix to its [[transpose]] is an involution. In the [[free semigroup]] generated by a [[nonempty set]] the [[Operation (mathematics)|operation]] which [[:wikt:reverse|reverse]]s the [[Order theory|order]] of the letters in a word is an involution.
| | Hi! <br>My name is Nichol and I'm a 19 years old boy from Netherlands.<br>xunjie 明示的にも黙示的に国際的なスーパーモデル、 |
| | 高品質の製品と完璧なアフターサービスを顧客に提供するために、 |
| | ファッションと追求の彼らの愛のユニークなスタイルと魅力の解釈で、 [http://athenavineyards.com/media/duvetica.php ���`�� �����] 彼女の背の高い数値の他に、 |
| | ほぼ万人の労力を投資して自分のビジネスを開始するのを助けるためにある広東省のようなよく知られた衣料品メーカー、 |
| | きれいなラインを立ち上げスタイルは、 [http://athenavineyards.com/media/duvetica.php �ޥå����ީ`�� ���`��] リトルプリンセス空想非常に非常に王女鵬ああ！愛をねたむません！お母さんが反映されている絹の上のセクションがあることを以前のようではない、 |
| | CINE21雑誌や写真撮影の衣類ブランドの推薦で、 |
| | オンラインショッピング市場を開始しました。[http://www.zgg.ch/data/video/shop/gaga.html �����ߥ�� �rӋ ��� ����] 日中は狭い引用し振幅ショック、 |
| | ハイエンドの衣服の処理の現在の傾向は明らかに制限されていない材料の無駄やアパレル物流業界が終了する生産段階で示して説明します。 |
| | 光沢のあるガードルまず私たちを見てみましょうダウン視線に沿って、 |
| | ナオミ·キャンベルとStellaTennantがあると判断しました！オリンピックをホストしているまた、 [http://www.schochauer.ch/_js/r/e/mall/watch/gaga/ �����ߥ�<br><br>Ӌ ����] |
|
| |
|
| ==Formal definition==
| | My homepage :: [http://bigapplefilmfestival.com/css/shop/Duvetica.php マックスマーラ ウィークエンド] |
| Let ''S'' be a semigroup. An involution in ''S'' is a [[unary operation]] * on ''S'' (or, a transformation * : ''S'' → ''S'', ''x'' ↦ ''x''*) satisfying the following conditions:
| |
| #For all ''x'' in ''S'', (''x''*)* = ''x''.
| |
| #For all ''x'', ''y'' in ''S'' we have (''xy'')* = ''y''*''x''*.
| |
| | |
| The semigroup ''S'' with the involution * is called a semigroup with involution.
| |
| | |
| ==Examples==
| |
| #If ''S'' is a [[commutative]] semigroup then the [[identity function|identity map]] of S is an involution.
| |
| #If ''S'' is a [[group (mathematics)|group]] then the inversion map * : ''S'' → ''S'' defined by ''x''* = ''x''<sup>−1</sup> is an involution.
| |
| #If ''S'' is an [[inverse semigroup]] then the inversion map is an involution which leaves the [[idempotent]]s [[Invariant (mathematics)|invariant]]. The inversion map is not necessarily the only map with this property in an inverse semigroup; there may well be other involutions that leave all idempotents invariant. A [[regular semigroup]] is an [[inverse semigroup]] if and only if it admits an involution under which each idempotent is an invariant.<ref>Munn, Lemma 1</ref>
| |
| #Underlying every [[C*-algebra]] is a *-semigroup. An important [[C*-algebra#Finite-dimensional C*-algebras|instance]] is the algebra ''M''<sub>''n''</sub>('''C''') of ''n''-by-''n'' [[matrix (mathematics)|matrices]] over '''[[Complex number|C]]''', with the [[conjugate transpose]] as involution.
| |
| #If ''X'' is a set, the set of all [[binary relation]]s on ''X'' is a *-semigroup with the * given by the [[inverse relation]], and the multiplication given by the usual [[composition of relations]].
| |
| #If X is a set, then the set of all finite sequences (or [[String (computer science)|strings]]) of members of X forms a [[free monoid]] under the operation of concatenation of sequences, with sequence reversal as an involution.<ref>[{{citation
| |
| | last1 = Crabb | first1 = M. J.
| |
| | last2 = McGregor | first2 = C. M.
| |
| | last3 = Munn | first3 = W. D.
| |
| | last4 = Wassermann | first4 = S.
| |
| | doi = 10.1017/S0308210500023179
| |
| | issue = 5
| |
| | journal = Proceedings of the Royal Society of Edinburgh
| |
| | mr = 1415814
| |
| | pages = 939–945
| |
| | title = On the algebra of a free monoid
| |
| | volume = 126
| |
| | year = 1996}}</ref>
| |
| | |
| ==Basic concepts and properties==
| |
| Certain basic concepts may be defined on *-semigroups in a way that parallels the notions stemming from a (von Neumann) regular element in a semigroup. A '''partial isometry''' is an element ''s'' when ''ss''*''s'' = ''s''; the set of partial isometries is usually abbreviated PI(''S''). A '''projection''' is an idempotent element ''e'' that is fixed by the involution, i.e. ''ee'' = ''e'' and ''e''* = ''e''. Every projection is a partial isometry, and for every partial isometry ''s'', ''s''*''s'' and ''ss''* are projections. If ''e'' and ''f'' are projections, then ''e'' = ''ef'' if and only if ''e'' = ''fe''.
| |
| | |
| Partial isometries can be [[partial order|partially ordered]] by ''s'' ≤ ''t'' [[if and only if]] ''s'' = ''ss''*''t'' and ''ss''* = ''ss''*''tt''*. Equivalently, ''s'' ≤ ''t'' if and only if ''s'' = ''et'' and ''e'' = ''ett''* for some projection ''e''. In a *-semigroup, PI(S) is an [[ordered groupoid]] with the partial product given by ''s''⋅''t'' = ''st'' if ''s''*''s'' = ''tt''*.
| |
| | |
| The [[partial isometry|partial isometries]] in a C*-algebra are exactly those defined in this section. In the case of ''M''<sub>''n''</sub>('''C''') more can be said. If ''E'' and ''F'' are projections, then ''E'' ≤ ''F'' if and only if [[Image (mathematics)|im]]''E'' ⊆ im''F''. For any two projection, if ''E'' ∩ ''F'' = ''V'', then the unique projection ''J'' with image ''V'' and kernel the [[orthogonal complement]] of ''V'' is the meet of ''E'' and ''F''. Since projections form a meet-[[semilattice]], the partial isometries on ''M''<sub>''n''</sub>('''C''') form an inverse semigroup with the product <math>A(A^*A\wedge BB^*)B</math>.<ref>Lawson p.120</ref>
| |
| | |
| ==*-regular semigroups==
| |
| A semigroup ''S'' with an involution * is called a '''*-regular semigroup''' if for every ''x'' in ''S'', ''x''* is ''H''-equivalent to some inverse of ''x'', where ''H'' is the [[Green's relations|Green’s relation]] ''H''. This defining property can be formulated in several equivalent ways. Another is to say that every [[Green's relations#The L.2C R.2C and J relations|''L''-class]] contains a projection. An axiomatic definition is the condition that for every ''x'' in ''S'' there exists an element ''x′'' such that ''x′xx′'' = ''x′'', ''xx′x'' = ''x'', (''xx′'')* = ''xx′'', (''x′x'')* = ''x′x''. [[Michael P. Drazin]] first proved that given ''x'', the element ''x′'' satisfying these axioms is unique. It is called the Moore–Penrose inverse of ''x''. This agrees with the classical definition of the [[Moore–Penrose inverse]] of a square matrix.
| |
| In the [[Matrix multiplication|multiplicative]] semigroup ''M''<sub>''n''</sub> ( ''C'' ) of square matrices of order ''n'', the map which assigns a matrix ''A'' to its [[Hermitian conjugate]] ''A''* is an involution. The semigroup ''M''<sub>''n''</sub> ( ''C'' ) is a *-regular semigroup with this involution. The Moore–Penrose inverse of A in this *-regular semigroup is the classical Moore–Penrose inverse of A.
| |
| | |
| ===P-systems===
| |
| An interesting question is to characterize when a [[regular semigroup]] is a *-regular semigroup. The following characterization was given by M. Yamada. Define a '''P-system''' F(S) as subset of the idempotents of S, denoted as usual by E(S). Using the usual notation V(''a'') for the inverses of ''a'', F(S) needs to satisfy the following axioms:
| |
| | |
| #For any ''a'' in S, there exists a unique a° in V(''a'') such that ''aa''° and ''a''°''a'' are in F(S)
| |
| #For any ''a'' in S, and b in F(S), a°ba is in F(S), where ° is the well-defined operation from the previous axiom
| |
| #For any ''a'', ''b'' in F(S), ab is in E(S); note: not necessarily in F(S)
| |
| | |
| A regular semigroup S is a *-regular semigroup, as defined by Nordahl & Scheiblich, if and only if it has a p-system F(S). In this case F(S) is the set of projections of S with respect to the operation ° defined by F(S). In an [[inverse semigroup]] the entire semilattice of idempotents is a p-system. Also, if a regular semigroup S has a p-system that is multiplicatively closed (i.e. subsemigroup), then S is an inverse semigroup. Thus, a p-system may be regarded as a generalization of the semilattice of idempotents of an inverse semigroup.
| |
| | |
| ==Free semigroup with involution==
| |
| Let <math>X,X^\dagger</math> be two [[disjoint sets]] in [[Bijection|bijective correspondence]] given by the [[Map (mathematics)|map]]
| |
| :<math>^\dagger:X\rightarrow X^\dagger</math>.
| |
| Denote by (here we use <math>\sqcup\,</math> instead of <math>\cup\,</math> to remind that the union is actually a [[disjoint union]])
| |
| :<math>Y=X\sqcup X^\dagger</math>
| |
| and by <math>Y^+\,</math> the [[free semigroup]] on <math>Y\,</math>. We can [[Function (mathematics)#Restrictions and extensions|extend]] the [[Map (mathematics)|map]] <math>{ }^\dagger\,</math> to a [[Map (mathematics)|map]]
| |
| :<math>{ }^\dagger:Y^+\rightarrow Y^+</math>
| |
| in the following way: given
| |
| :<math>w = w_1w_2 \cdots w_k \in Y^+</math> for some letters <math>w_i\in Y</math>
| |
| then we define
| |
| :<math>w^\dagger=w_k^\dagger w_{k-1}^\dagger \cdots w_{2}^\dagger w_{1}^\dagger.</math>
| |
| This [[Map (mathematics)|map]] is an [[#Formal definition|involution]] on the semigroup <math>Y^+\,</math>. This is the only way to [[Function (mathematics)#Restrictions and extensions|extend]] the map <math>{ }^\dagger\,</math> from <math>X\,</math> to <math>X^\dagger\,</math>, to an [[#Formal definition|involution]] on <math>Y^+\,</math>.
| |
| | |
| Thus, the semigroup <math>(X\sqcup X^\dagger)^+</math> with the map <math>{ }^\dagger\,</math> is a semigroup with involution. Moreover, it is the '''free semigroup with involution''' on <math>X\,</math> in the sense that it solves the following [[universal algebra|universal problem]]: given a semigroup with [[#Formal definition|involution]] <math>S\,</math> and a map
| |
| :<math>\Phi:X\rightarrow S</math>,
| |
| a semigroup [[homomorphism]]
| |
| :<math>\overline\Phi:(X\sqcup X^\dagger)^+\rightarrow S</math>
| |
| exists such that
| |
| :<math>\Phi = \iota \circ \overline\Phi</math>
| |
| where
| |
| :<math>\iota : X \rightarrow (X\sqcup X^\dagger)^+</math>
| |
| is the [[inclusion map]] and [[composition of functions]] is taken in the [[Function composition#Alternative notations|diagram order]].
| |
| It is well known from [[universal algebra]] that <math>(X\sqcup X^\ddagger)^+</math> is unique up to [[isomorphism]]s.
| |
| | |
| If we use <math>Y^*\,</math> instead of <math>Y^+\,</math>, where
| |
| :<math>Y^*=Y^+\cup\{\varepsilon\}</math> | |
| where <math>\varepsilon\,</math> is the [[empty word]] (the [[Identity element|identity]] of the [[monoid]] <math>Y^*\,</math>), we obtain a [[monoid]] with [[#Formal definition|involution]] <math>(X\sqcup X^\dagger)^*</math> that is the '''free monoid with involution''' on <math>X\,</math>.
| |
| | |
| ==See also==
| |
| * [[Dagger category]]
| |
| * [[Special classes of semigroups]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| *Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". [[World Scientific]] ISBN 981-02-3316-7
| |
| *D J Foulis (1958). ''Involution Semigroups'', Ph.D. Thesis, Tulane University, New Orleans, LA. [http://www.math.umass.edu/~foulis/publ.txt Publications of D.J. Foulis] (Accessed on 5 May 2009)
| |
| * W.D. Munn, ''Special Involutions'', in A.H. Clifford, K.H. Hofmann, M.W. Mislove, ''Semigroup theory and its applications: proceedings of the 1994 conference commemorating the work of Alfred H. Clifford'', Cambridge University Press, 1996, ISBN 0521576695. This is a recent survey article on semigroup with (special) involution
| |
| * Drazin, M.P., ''Regular semigroups with involution'', Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46
| |
| * Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum, 16(1978), 369–377.
| |
| * Miyuki Yamada, ''P-systems in regular semigroups'', Semigroup Forum, 24(1), December 1982, pp. 173–187
| |
| | |
| *{{PlanetMath attribution|id=8283|title=Free semigroup with involution}}
| |
| {{Use dmy dates|date=September 2010}}
| |
| | |
| {{DEFAULTSORT:Semigroup With Involution}}
| |
| [[Category:Algebraic structures]]
| |
| [[Category:Semigroup theory]]
| |