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 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: [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, is