Four-spiral semigroup

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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.[1][2] It has several interesting properties: it is one of the most important examples of bi-simple but not completely-simple semigroups;[3] it is also an important example of a fundamental regular semigroup;[2] it is an indispensable building block of bisimple, idempotent-generated regular semigroups.[2] A certain semigroup, called double four-spiral semigroup, generated by five idempotent elements has also been studied along with the four-spiral semigroup.[4][2]

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:[2]

  • 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 ωl a, where ωl is a biorder relation defined by Nambooripad. The diagram below summarises the various relations among a, b, c, d:

[math]\displaystyle{ \begin{matrix} & & \mathcal{R} & & \\ & a & \longleftrightarrow & b & \\ \omega^l & \Big \uparrow & & \Big \updownarrow & \mathcal{L} \\ & d & \longleftrightarrow & c & \\ & & \mathcal{R} & & \end{matrix} }[/math]

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.[4]

General elements

Every element of Sp4 can be written uniquely in one of the following forms:[2]

[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 = ABCDE 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 DE is a nonregular semigroup.

Idempotent elements

The set of idempotents of Sp4,[5] is {an, bn, cn, dn : n = 0, 1, 2, ...} where, a0 = a, b0 = b, c0 = c, d0 = d, and for n = 0, 1, 2, ....,

an+1 = a(ca)n(db)nd
bn+1 = a(ca)n(db)n+1
cn+1 = (ca)n+1(db)n+1
dn+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:

EA = { an : n = 0,1,2, ... }
EB = { bn : n = 0,1,2, ... }
EC = { cn : n = 0,1,2, ... }
ED = { dn : 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

[math]\displaystyle{ (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} }[/math]

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.[2]

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:[2][4]

  • 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 = ωl ∩ ωr.

References

  1. Byleen, K. (1977). The Structure of Regular and Inverse Semigroups, Doctoral Dissertation. University of Nebraska. 
  2. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Pierre Antoine Grillet (1996). "On the fundamental double four-spiral semigroup". Bulletin of the Belgian Mathematical Society 3: 201 − 208. 
  3. L.N. Shevrin (originator). "Simple semi-group". Encyclopedia of Mathematics. http://www.encyclopediaofmath.org/index.php?title=Simple_semi-group&oldid=18138. Retrieved 25 January 2014. 
  4. 4.0 4.1 4.2 Meakin, John; K. Byleen; F. Pastijn (1980). "The double four-spiral semigroup". Simon Stevin 54: 75 & minus 105. 
  5. Karl Byleen; John Meakin; Francis Pastjin (1978). "The Fundamental Four-Spiral Semigroup". Journal of Algebra 54: 6 − 26. doi:10.1016/0021-8693(78)90018-2.