{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. 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 = ωl ∩ ωr. References Semigroup theory