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.[2]
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]
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:
\begin{matrix} &&l{R}&&\\ &a&\longleftrightarrow&b&\\ \omegal& \uparrow&& \updownarrow&l{L}\\ &d&\longleftrightarrow&c&\\ &&l{R}&& \end{matrix}
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 =
B =
C =
D =
E =
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,[4] is 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 =
EB =
EC =
ED =
Let S be the set of all quadruples (r, x, y, s) where r, s, ∈ 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)&ifs=0,t=1\\ (r,x-y+max(y,z),max(y,z)-z+w,u)&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.[2]
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]
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.