Four-spiral Semigroup

In mathematics, the four-spiral semigroup is a special semigroup generated by four idempotent elements. This special semigroup was first studied by Karl Byleen in a doctoral dissertation submitted to the University of Nebraska in 1977. It has several interesting properties: it is one of the most important examples of bi-simple but not completely-simple semigroups; it is also an important example of a fundamental regular semigroup; it is an indispensable building block of bisimple, idempotent-generated regular semigroups. A certain semigroup, called double four-spiral semigroup, generated by five idempotent elements has also been studied along with the four-spiral semigroup.

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:

:* ''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'' ω^l ''a'', where ω^l is a biorder relation defined by Nambooripad. The diagram below summarises the various relations among ''a'', ''b'', ''c'', ''d'':

$\begin & & \mathcal & & \\ & a & \longleftrightarrow & b & \\ \omega^l & \Big \uparrow & & \Big \updownarrow & \mathcal \\ & d & \longleftrightarrow & c & \\ & & \mathcal & & \end$

Elements of the four-spiral semigroup

General elements

Every element of ''Sp''₄ can be written uniquely in one of the following forms:

:: 'c''(''ac'')^''m'' :: '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 ''Sp''₄ = ''A'' ∪ ''B'' ∪ ''C'' ∪ ''D'' ∪ ''E'' where
:: ''A'' =
:: ''B'' =
:: ''C'' =
:: ''D'' =
:: ''E'' =

The sets ''A'', ''B'', ''C'', ''D'' are bicyclic semigroups, ''E'' is an infinite cyclic semigroup and the subsemigroup ''D'' ∪ ''E'' is a nonregular semigroup.

Idempotent elements

The set of idempotents of ''Sp''₄, is 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''(''ca'')^''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'' =
:: ''E''_''B'' =
:: ''E''_''C'' =
:: ''E''_''D'' =

Four-spiral semigroup as a Rees-matrix semigroup

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

$(r, x, y, s) * (t, z, w, u) = \begin (r, x-y + \max(y , z + 1), \max(y - 1, z) - z + w, u) & \text s = 0, t = 1\\ (r, x - y+ \max(y, z), \max(y, z) - z + w, u)&\text \end$

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''.

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:

:*''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 idempotents, 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 biorder relation defined as ω = ω^l ∩ ω^r.

References

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Semigroup theory