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词条 Four-spiral semigroup
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  1. Definition

  2. Elements of the four-spiral semigroup

      General elements    Idempotent elements  

  3. Four-spiral semigroup as a Rees-matrix semigroup

  4. Properties

  5. Double four-spiral semigroup

  6. References

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.[6][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:

Elements of the four-spiral semigroup

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 = 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,[4] 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

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][5]

  • 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. ^{{cite book|last=Byleen, K.|title=The Structure of Regular and Inverse Semigroups, Doctoral Dissertation|year=1977|publisher=University of Nebraska}}
2. ^1 2 3 4 5 6 7 {{cite journal|last=Pierre Antoine Grillet|title=On the fundamental double four-spiral semigroup|journal=Bulletin of the Belgian Mathematical Society|year=1996|volume=3|pages=201 − 208}}
3. ^{{cite web|last=L.N. Shevrin (originator)|title=Simple semi-group|url=http://www.encyclopediaofmath.org/index.php?title=Simple_semi-group&oldid=18138|work=Encyclopedia of Mathematics|accessdate=25 January 2014}}
4. ^{{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}}
5. ^1 2 {{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}}

1 : Semigroup theory
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