# 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)$^m$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 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)$^n$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$_A$ = { 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$^r$.