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:

• a² = a, b² = b, c² = c, d² = 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

The spiral structure of idempotents in the four-spiral semigroup Sp4. In this diagram, elements in the same row are R-related, elements in the same column are L-related, and the order proceeds down the four diagonals (away from the center).

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.

General elements

Every element of Sp₄ 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 Sp₄ has a partition Sp₄ = 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⁺¹d  : 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 Sp₄, is {a_n, b_n, c_n, d_n : n = 0, 1, 2, ...} where, a₀ = a, b₀ = b, c₀ = c, d₀ = 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+ld

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 Sp₄ is isomorphic to S.

Properties

• By definition itself, the four-spiral semigroup is an idempotent generated semigroup (Sp₄ is generated by the four idempotents a, b. c, d.)
• The four-spiral semigroup is a fundamental semigroup, that is, the only congruence on Sp₄ which is contained in the Green's relation H in Sp₄ is the equality relation.

Double four-spiral semigroup

The fundamental double four-spiral semigroup, denoted by DSp₄, is the semigroup generated by five elements a, b, c, d, e satisfying the following conditions:

• a² = a, b² = b, c² = c, d² = d, e² = 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.