|
|
| Line 1: |
Line 1: |
| A '''regular semigroup''' is a [[semigroup]] ''S'' in which every element is '''regular''', i.e., for each element ''a'', there exists an element ''x'' such that ''axa'' = ''a''.<ref>Howie 1995 : 54.</ref> Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via [[Green's relations]].<ref>Howie 2002.</ref>
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is where my primary income arrives from but I've usually wanted my personal company. Mississippi is where his home is. To climb is some thing I truly enjoy performing.<br><br>Also visit my web blog - online psychic chat ([http://Cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 Cpacs.org]) |
| | |
| == Origins ==
| |
| Regular semigroups were introduced by [[J. A. Green]] in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which [[Green's relations]] were introduced. The concept of ''regularity'' in a semigroup was adapted from an analogous condition for [[Ring (mathematics)|rings]], already considered by [[J. von Neumann]].<ref>von Neumann 1936.</ref> It was his study of regular semigroups which led Green to define his celebrated [[Green's relations|relations]]. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to [[semigroup]]s was first made by [[David Rees (mathematician)|David Rees]].
| |
| | |
| == The basics ==
| |
| There are two equivalent ways in which to define a regular semigroup ''S'':
| |
| :(1) for each ''a'' in ''S'', there is an ''x'' in ''S'', which is called a '''pseudoinverse''',<ref>Klip, Knauer and Mikhalev : p. 33</ref> with ''axa'' = ''a'';
| |
| :(2) every element ''a'' has at least one '''inverse''' ''b'', in the sense that ''aba'' = ''a'' and ''bab'' = ''b''.
| |
| To see the equivalence of these definitions, first suppose that ''S'' is defined by (2). Then ''b'' serves as the required ''x'' in (1). Conversely, if ''S'' is defined by (1), then ''xax'' is an inverse for ''a'', since ''a''(''xax'')''a'' = ''axa''(''xa'') = ''axa'' = ''a'' and (''xax'')''a''(''xax'') = ''x''(''axa'')(''xax'') = ''x''(''axa'')''x'' = ''xax''.<ref>Clifford and Preston 1961 : Lemma 1.14.</ref>
| |
| | |
| The set of inverses (in the above sense) of an element ''a'' in an arbitrary [[semigroup]] ''S'' is denoted by ''V''(''a'').<ref>Howie 1995 : p. 52.</ref> Thus, another way of expressing definition (2) above is to say that in a regular semigroup, ''V''(''a'') is nonempty, for every ''a'' in ''S''. The product of any element ''a'' with any ''b'' in ''V''(''a'') is always [[idempotent]]: ''abab'' = ''ab'', since ''aba'' = ''a''.<ref>Clifford and Preston 1961 : p. 26.</ref>
| |
| | |
| A regular semigroup in which [[idempotent]]s commute is an [[inverse semigroup]], that is, every element has a ''unique'' inverse. To see this, let ''S'' be a regular semigroup in which [[idempotent]]s commute. Then every element of ''S'' has at least one inverse. Suppose that ''a'' in ''S'' has two inverses ''b'' and ''c'', i.e.,
| |
| :''aba'' = ''a'', ''bab'' = ''b'', ''aca'' = ''a'' and ''cac'' = ''c''. Also ''ab'', ''ba'', ''ac'' and ''ca'' are idempotents as above.
| |
| Then
| |
| :''b'' = ''bab'' = ''b''(''aca'')''b'' = ''bac''(''a'')''b =''bac''(''aca'')''b = ''bac''(''ac'')(''ab'') = ''bac''(''ab'')(''ac'') = ''ba''(''ca'')''bac'' = ''ca''(''ba'')''bac'' = ''c''(''aba'')''bac'' = ''cabac'' = ''cac'' = ''c''.
| |
| So, by commuting the pairs of [[idempotent]]s ''ab'' & ''ac'' and ''ba'' & ''ca'', the inverse of ''a'' is shown to be unique. Conversely, it can be shown that any [[inverse semigroup]] is a regular semigroup in which [[idempotent]]s commute.<ref name="Howie 1995 : Theorem 5.1.1">Howie 1995 : Theorem 5.1.1.</ref>
| |
| | |
| The existence of a unique pseudoinverse implies the existence of a unique inverse, but the opposite is not true. For example, in the [[symmetric inverse semigroup]], the empty transformation Ø does not have a unique pseudoinverse, because Ø = Ø''f''Ø for any transformation ''f''. The inverse of Ø is unique however, because only one ''f'' satisfies the additional constraint that ''f'' = Ø''f''Ø, namely ''f'' = Ø. This remark holds more generally in any semigroup with zero. Furthermore, if every element has a unique pseudoinverse, then the semigroup is a [[group (mathematics)|group]], and the unique pseudoinverse of an element coincides with the group inverse.<ref>Proof: http://planetmath.org/?op=getobj&from=objects&id=6391</ref>
| |
| | |
| '''Theorem.''' The homomorphic image of a regular semigroup is regular.<ref>Howie 1995 : Lemma 2.4.4.</ref>
| |
| | |
| Examples of regular semigroups:
| |
| *Every [[group (mathematics)|group]] is regular.
| |
| *Every [[inverse semigroup]] is regular.
| |
| *Every [[band (mathematics)|band]] (idempotent semigroup) is regular in the sense of this article, though this is not what is meant by a [[band (mathematics)#Regular bands|regular band]].
| |
| *The [[bicyclic semigroup]] is regular.
| |
| *Any [[transformation semigroup|full transformation semigroup]] is regular.
| |
| *A [[Rees matrix semigroup]] is regular.
| |
| | |
| == Green's relations ==
| |
| Recall that the [[principal ideal]]s of a semigroup ''S'' are defined in terms of ''S''<sup>1</sup>, the ''[[semigroup]] with identity adjoined''; this is to ensure that an element ''a'' belongs to the principal right, left and two-sided [[ideal (ring theory)|ideal]]s which it generates. In a regular semigroup ''S'', however, an element ''a'' = ''axa'' automatically belongs to these ideals, without recourse to adjoining an identity. [[Green's relations]] can therefore be redefined for regular semigroups as follows:
| |
| :<math>a\,\mathcal{L}\,b</math> if, and only if, ''Sa'' = ''Sb'';
| |
| :<math>a\,\mathcal{R}\,b</math> if, and only if, ''aS'' = ''bS'';
| |
| :<math>a\,\mathcal{J}\,b</math> if, and only if, ''SaS'' = ''SbS''.<ref>Howie 1995 : 55.</ref>
| |
| | |
| In a regular semigroup ''S'', every <math>\mathcal{L}</math>- and <math>\mathcal{R}</math>-class contains at least one [[idempotent]]. If ''a'' is any element of ''S'' and α is any inverse for ''a'', then ''a'' is <math>\mathcal{L}</math>-related to ''αa'' and <math>\mathcal{R}</math>-related to ''aα''.<ref>Clifford and Preston 1961 : Lemma 1.13.</ref>
| |
| | |
| '''Theorem.''' Let ''S'' be a regular semigroup, and let ''a'' and ''b'' be elements of ''S''. Then
| |
| *<math>a\,\mathcal{L}\,b</math> if, and only if, there exist α in ''V''(''a'') and β in ''V''(''b'') such that α''a'' = β''b'';
| |
| *<math>a\,\mathcal{R}\,b</math> if, and only if, there exist α in ''V''(''a'') and β in ''V''(''b'') such that ''a''α = ''b''β.<ref>Howie 1995 : Proposition 2.4.1.</ref>
| |
| | |
| If ''S'' is an [[inverse semigroup]], then the idempotent in each <math>\mathcal{L}</math>- and <math>\mathcal{R}</math>-class is unique.<ref name="Howie 1995 : Theorem 5.1.1"/>
| |
| | |
| == Special classes of regular semigroups ==
| |
| Some special classes of regular semigroups are:<ref>Howie 1995 : Section 2.4 & Chapter 6.</ref>
| |
| *''Locally inverse semigroups'': a regular semigroup ''S'' is '''locally inverse''' if ''eSe'' is an inverse semigroup, for each [[idempotent]] ''e''.
| |
| *''Orthodox semigroups'': a regular semigroup ''S'' is '''orthodox''' if its subset of [[idempotent]]s forms a subsemigroup.
| |
| *''Generalised inverse semigroups'': a regular semigroup ''S'' is called a '''generalised inverse semigroup''' if its [[idempotent]]s form a normal band, i.e., ''xyzx'' = ''xzyx'', for all [[idempotent]]s ''x'', ''y'', ''z''.
| |
| The [[class (set theory)|class]] of generalised inverse semigroups is the [[intersection (set theory)|intersection]] of the class of locally inverse semigroups and the class of orthodox semigroups.<ref>Howie 1995 : 222.</ref>
| |
| | |
| == See also ==
| |
| | |
| *[[Biordered set]]
| |
| *[[Special classes of semigroups]]
| |
| *[[Nambooripad order]]
| |
| | |
| == Notes ==
| |
| {{reflist|2}}
| |
| | |
| == References ==
| |
| *A. H. Clifford and G. B. Preston, ''The Algebraic Theory of Semigroups'', Volume 1, Mathematical Surveys of the American Mathematical Society, No. 7, Providence, R.I., 1961.
| |
| *J. M. Howie, ''Fundamentals of Semigroup Theory'', Clarendon Press, Oxford, 1995.
| |
| *M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.
| |
| *{{cite journal | author=J. A. Green | title=On the structure of semigroups | journal=Annals of Mathematics. Second Series | year=1951 | volume=54 | pages=163–172 | doi=10.2307/1969317 | issue=1 | publisher=Annals of Mathematics | jstor=1969317}}
| |
| *J. M. Howie, Semigroups, past, present and future, ''Proceedings of the International Conference on Algebra and Its Applications'', 2002, 6–20.
| |
| *{{cite journal | author=J. von Neumann | title=On regular rings | journal=Proceedings of the National Academy of Sciences of the USA | year=1936 | volume=22 | pages=707–713 | doi=10.1073/pnas.22.12.707 | pmid=16577757 | issue=12 | pmc=1076849}}
| |
| | |
| [[Category:Semigroup theory]]
| |
I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is where my primary income arrives from but I've usually wanted my personal company. Mississippi is where his home is. To climb is some thing I truly enjoy performing.
Also visit my web blog - online psychic chat (Cpacs.org)