Four-spiral semigroup

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}

File:Spiral_Structure_of_idempotents_in_Sp4.png s 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).]]

File:Four_Spiral_Semigroup_02.png

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.

The set of idempotents of Sp₄,{{cite journal|last=Karl Byleen|author2=John Meakin |author3=Francis Pastjin |title=The Fundamental Four-Spiral Semigroup|journal=Journal of Algebra|year=1978|volume=54|pages=6 − 26|doi=10.1016/0021-8693(78)90018-2|doi-access=free}} 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, ... }

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.

• 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.

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:{{cite journal|last=Meakin|first=John |author2=K. Byleen |author3=F. Pastijn|title=The double four-spiral semigroup|journal=Simon Stevin|year=1980|volume=54|pages=75 & minus 105}}

:*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.

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Category:Semigroup theory

Category:4 (number)

Category:Spirals