Four-spiral semigroup

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:
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":
formula_1
Elements of the four-spiral semigroup.
General elements.
Every element of "Sp"4 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 "Sp"4 has a partition "Sp"4 = "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 "Sp"4, is {"a""n", "b""n", "c""n", "d""n" : "n" = 0, 1, 2, ...} where, "a"0 = "a", "b"0 = "b", "c"0 = "c", "d"0 = "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
formula_2
The set "S" with this operation is a Rees matrix semigroup over the bicyclic semigroup, and the four-spiral semigroup "Sp"4 is isomorphic to "S".
Double four-spiral semigroup.
The fundamental double four-spiral semigroup, denoted by "DSp"4, is the semigroup generated by five elements "a", "b", "c", "d", "e" satisfying the following conditions:
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.