Four-spiral semigroup

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.

[4][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.

Every element of Sp4 can be written uniquely in one of the following forms:[2] 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 set of idempotents of Sp4,[5] is {an, bn, cn, dn : 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: 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