Four spiral semigroup - Alchetron, The Free Social Encyclopedia

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:

R a ⟷ b ω l ↑ ↕ L d ⟷ c R

General elements

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

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

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 ,....,

The sets of idempotents in the subsemigroups A, B, C, D (there are no idempotents in the subsemigoup E) are respectively:

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 ) = { ( r , x − y + max ( y , z + 1 ) , max ( y − 1 , z ) − z + w , u ) if s = 0 , t = 1 ( r , x − y + max ( y , z ) , max ( y , z ) − z + w , u ) otherwise.

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.

References

Four-spiral semigroup Wikipedia

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