On the fundamental double four-spiral semigroup

We give a new description of the fundamental double four-spiral semigroup. The fundamental four-spiral semigroup Sp4 and the fundamental double fourspiral semigroup DSp4 were introduced in [1], [3], and [4]. These semigroups are interesting examples of fundamental regular semigroups, and are indispensable building blocks of bisimple, idempotent-generated regular semigroups. Their basic properties are recalled in parts 1 and 2 of this note. In part 3 we give an alternate construction of DSp4 in terms of the free semigroup on two generators, as a set of quadruples with a simple, bicyclic-like multiplication. This permits shorter proofs and easier access to the main properties of DSp4: descriptions of DSp4/L and DSp4/R (part 4); reduced form of the elements (part 5); and the property of congruences C 6⊆ L that DSp4/C is completely simple (part 6). 1. Recall that Sp4 is the semigroup Sp4 ∼= 〈a, b, c, d; a = a, b = b, c = c, d = d, a = ba, b = ab, b = bc, c = cb, c = dc, d = cd, d = da〉 generated by four idempotents a, b, c, d such that aR b L c R d ≤L a. (We denote Green’s left preorder x ∈ Sy by x ≤L y). It is shown in [3] that every element of Sp4 can be written uniquely in reduced form [c](ac)[a], [d](bd)[b], [c](ac)ad(bd)[b], Received by the editors March 1995. Communicated by J. Thas. 1991 Mathematics Subject Classification : 20M17.