en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)

2014-01-27T06:09:23Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)

New page

In [[mathematics]], the '''four-spiral semigroup''' is a special [[semigroup]] generated by four [[idempotent]] elements. This special spemigroup was first studied by Byleen K in a doctoral dissertation submitted to [[University of Nebraska]] in 1977.<ref>{{cite book|last=Byleen, K.|title=''The Structure of Regular and Inverse Semigroups'', Doctoral Dissertation|year=1977|publisher=University of Nebraska}}</ref><ref name="Grillet">{{cite journal|last=Pierre Antoine Grillet|title=On the fundamental double four-spiral semigroup|journal=Bulletin of Belgian Mathematical Society|year=1996|volume=3|pages=201 − 208}}</ref> It has several interesting properties: it is one of the most important examples of bi-simple but not completely-simple semigroups;<ref>{{cite web|last=L.N. Shevrin (originator)|title=Simple semi-group|url=http://www.encyclopediaofmath.org/index.php?title=Simple_semi-group&oldid=18138|work=Encyclopedia of Mathematics|accessdate=25 January 2014}}</ref> it is also an important example of a fundamental [[regular semigroup]];<ref name="Grillet"/> it is an indispensable building block of bisimple, idempotent-generated regular semigroups.<ref name="Grillet"/> A certain semigroup, called '''double four-spiral semigroup''', generated by five idempotent elements has also been studied along with the four-spiral semigroup.<ref name="Meakin"/><ref name="Grillet"/>

==Definition==
The four-spiral semigroup, denoted by ''Sp4'', is the [[free semigroup]] generated by four elements ''a'', ''b'', ''c'', and ''d'' satisfying the following eleven conditions:<ref name="Grillet"/>

:* ''a''2 = ''a'', ''b''2 = ''b'', ''c''2 = ''c'', ''d''2 = ''d''.
:* ''ab'' = ''b'', ''ba'' = ''a'', ''bc'' = ''b'', ''cb'' = ''c'', ''cd'' = ''d'', ''dc'' = ''c''.
:* ''da'' = ''d''.

The first set of conditions imply that the elements ''a'', ''b'', ''c'', ''d'' are idempotents. The second set of conditions imply that ''a R b L c R d'' where ''R'' and ''L'' are the [[Green's relations]] in a semigroup. The lone condition in the third set can be written as ''d'' ωl ''a'', where ωl is a [[biordered set|biorder relation]] defined by [[K. S. S. Nambooripad|Nambooripad]]. The diagram belo summarises the various relations among ''a'', ''b'', ''c'', ''d'':

<math>
\begin{matrix}
& & \mathcal{R} & & \\
& a & \longleftrightarrow & b & \\
\omega^l & \Big \uparrow & & \Big \updownarrow & \mathcal{L} \\
& d & \longleftrightarrow & c & \\
& & \mathcal{R} & &
\end{matrix}
</math>

==Elements of the four-spiral semigroup==

[[File:Spiral_Structure_of_idempotents_in_Sp4.png |right|375px|thumb|The spiral structure of [[idempotent]]s in the four-spiral semigroup Sp4. In this diagram, elements in the same row are [[Green's relations|R-related]], elements in the same column are [[Green's relations|L-related]], and the order proceeds down the four diagonals (away from the center).]]

[[File:Four_Spiral_Semigroup_02.png|right|375px|thumb|The structure of the four-spiral semigroup Sp4. The set of idempotents (red coloured points) and the subsemigroups A, B, C, D, E are shown.<ref name="Meakin"/>]]

=== General elements ===
Every element of ''Sp''4 can be written uniquely in one of the following forms:<ref name="Grillet"/>

:: [''c''] (''ac'')''m'' [a]
:: [''d''] (''bd'')''n'' [''b'']
:: [''c''] (''ac'')''m'' ''ad'' (''bd'')''n'' [''b'']
where ''m'' and ''n'' are non-negative integers and terms in square brackets may be omitted as long as the remaining product is not empty. The forms of these elements imply that ''Sp''4 has a [[partition of a set|partition]] ''Sp''4 = ''A'' ∪ ''B'' ∪ ''C'' ∪ ''D'' ∪ ''E'' where
:: ''A'' = { ''a''(''ca'')''n'', (''bd'')''n''+1, ''a''(''ca'')''m''''d''(''bd'')''n'' : ''m'', ''n'' non-negative integers }
:: ''B'' = { (''ac'')''n''+1, ''b''(''db'')''n'', ''a''(''ca'')''m''(''db'') ''n''+1 : ''m'', ''n'' non-negative integers }
:: ''C'' = { ''c''(''ac'')''m'', (''db'')''n''+1, (''ca'')''m''+1(''db'')''n''+1 : ''m'', ''n'' non-negative integers }
:: ''D'' = { ''d''(''bd'')''n'', (''ca'')''m''+1(''db'')''n''+1''d'' : ''m'', ''n'' non-negative integers }
:: ''E'' = { (''ca'')''m'' : ''m'' positive integer }

The sets ''A'', ''B'', ''C'', ''D'' are [[bicyclic semigroup]]s, ''E'' is an infinite [[cyclic semigroup]] and the subsemigroup ''D'' ∪ ''E'' is a [[regular semigroup|nonregular semigroup]].

=== Idempotent elements ===
The set of idempotents of ''Sp''4,<ref>{{cite journal|last=Karl Byleen|coauthors=John Meakin, Francis Pastjin|title=The Fundamental Four-Spiral Semigroup|journal=Journal of Algebra|year=1978|volume=54|pages=6 − 26}}</ref> is {''a''''n'', ''b''''n'', ''c''''n'', ''d''''n'' : ''n'' = 0, 1, 2 ,...} where, ''a''0 = ''a'', ''b''0 = ''b'', ''c''0 = ''c'', ''d''0 = ''d'', and for ''n'' = 0, 1, 2 ,....,
:: ''a''''n''+1 = ''a''(''ca'')''n''(''db'')''n''''d''
:: ''b''''n''+1 = ''a''(''cu'')''n''(''db'')''n''+1
:: ''c''''n''+1 = (''ca'')''n''+1(''db'')''n''+1
:: ''d''''n''+1 = (''ca'')''n''+1(''db'')''n''+l''d''

The sets of idempotents in the subsemigroups ''A'', ''B'', ''C'', ''D'' (there are no idempotents in the subsemigoup ''E'') are respectively:

:: ''E''''A'' = { ''a''''n'' : ''n'' = 0,1,2, ... }
:: ''E''''B'' = { ''b''''n'' : ''n'' = 0,1,2, ... }
:: ''E''''C'' = { ''c''''n'' : ''n'' = 0,1,2, ... }
:: ''E''''D'' = { ''d''''n'' : ''n'' = 0,1,2, ... }

==Four-spiral semigroup as a Rees-matrix semigroup==

Let ''S'' be the set of all qudruples (''r'', ''x'', ''y'', ''s'') where ''r'', ''s'', ∈ { 0, 1 } and ''x'' and ''y'' are nonnegative integers and define a binary operation in ''S'' by

<math>
(r, x, y, s) * (t, z, w, u) =
\begin{cases}
(r, x-y + \max(y , z + 1), \max(y - 1, z) - z + w, u) & \text{if } s = 0, t = 1\\
(r, x - y+ \max(y, z), \max(y, z) - z + w, u)&\text{otherwise.}
\end{cases}
</math>

The set ''S'' with this operation is a Rees matrix semigroup over the [[bicyclic semigroup]], and the four-spiral semigroup ''Sp''4 is isomorphic to ''S''.<ref name="Grillet"/>

==Properties==
*By definition itself, the four-spiral semigroup is an ''idempotent generated semigroup'' (''Sp''4 is generated by the four idempotents ''a'', ''b''. ''c'', ''d''.)
*The four-spiral semigroup is a fundamental semigroup, that is, the only congruence on ''Sp''4 which is contained in the Green's relation ''H'' in ''Sp''4 is the equality relation.

==Double four-spiral semigroup==

The '''fundamental double four-spiral semigroup''', denoted by ''DSp''4, is the semigroup generated by five elements ''a'', ''b'', ''c'', ''d'', ''e'' satisfying the following conditions:<ref name="Grillet"/><ref name="Meakin">{{cite journal|last=Meakin|first=John|coauthors=K. Byleen and F. Pastijn|title=The double four-spiral semigroup|journal=Simon Stevin|year=1980|volume=54|pages=75 & minus 105}}</ref>

:*''a''2 = ''a'', ''b''2 = ''b'', ''c''2 = ''c'', ''d''2 = ''d'', ''e''2 = ''e''
:*''ab'' = ''b'', ''ba'' = ''a'', ''bc'' = ''b'', ''cb'' = ''c'', ''cd'' = ''d'', ''dc'' = ''c'', ''de'' = ''d'', ''ed'' = ''e''
:*''ae'' = ''e'', ''ea'' = ''e''

The first set of conditions imply that the elements ''a'', ''b'', ''c'', ''d'', ''e'' are idempotents. The second set of conditions state the Green's relations among these idmpotents, namely, ''a R b L c R d L e''. The two conditions in the third set imply that ''e'' ω ''a'' where ω is the [[biordered set|biorder relation]] defined as ω = ωl ∩ ωr.

==References==
{{reflist}}

[[Category:Semigroup theory]]

Maximum disjoint set - Revision history

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)