Four-spiral semigroup
# 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 $Sp_4,$ 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:

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

## Elements of the four-spiral semigroup

### General elements

Every element of $Sp_4$ can be written uniquely in one of the following forms:

       [c] $(ac)^m$ [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 $Sp_4$ = A ∪ B ∪ C ∪ D ∪ E where

       A = { $a(ca)^n,$ $(bd)^{n+1},$ $a(ca)^md(bd)^n$ : 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 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 ${a_n,$ $b_n,$ $c_n,$ $d_n$ : n = 0, 1, 2, ...} where, $a_0$ = a, $b_0$ = b, $c_0$ = c, $d_0$ = d, and for n = 0, 1, 2, ....,

       $a_{n+1}$ = $a(ca)^n(db)^nd$
       $b_{n+1}$ = $a(ca)^n(db)^{n+1}$
       $c_{n+1}$ = $(ca)^{n+1}(db)^{n+1}$
       $d_{n+1}$ = $(ca)^{n+1}(db)^{n+l}d$

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

       $E_A$ = { $a_n$ : 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 quadruples (r, x, y, s) where r, s, ∈ { 0, 1 } and x and y are nonnegative integers and define a binary operation in S by

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

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 = $ω^l$ ∩ $ω^r.$

