Periodic Elements of the Free Idempotent Generated Semigroup on a Biordered Set

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup. The biordered set of a semigroup S is the set of idempotents of S considered as a partial groupoid with respect to the restriction of the multiplication of S to those pairs (e, f) of idempotents such that ef = e , ef = f , fe = e or fe = f . Nambooripad [6] who has initiated an axiomatic approach to biordered sets has defined an abstract biordered set as a partial groupoid satisfying certain second order axioms. The first author [3] has confirmed the adequacy of Nambooripad’s axiomatization by showing that each abstract biordered set is in fact the biordered set of a suitable semigroup. Namely, if 〈E, ◦〉 is an abstract biordered set, denote by IG(E) the semigroup with presentation IG(E) = {E | ef = e ◦ f whenever e ◦ f is defined in E}. The semigroup IG(E) is called the free idempotent generated semigroup on E . In [3] it has been shown that the biordered set of IG(E) coincides with the initial biordered set 〈E, ◦〉 (see Lemma 2 below for a precise formulation of this result). The structure of the free idempotent generated semigroup on a biordered set is not yet well understood. It was conjectured that subgroups of such a semigroup should be free. Though confirmed for some partial cases (see [5, 7, 8, 9]), this conjecture has been recently disproved by Brittenham, Margolis, and Meakin [1] who have found a biordered set 〈E, ◦〉 such that the semigroup IG(E) has the free abelian group of rank 2 among its subgroups. Moreover, in the subsequent paper [2] the same authors have proved that if F is any field, and E3(F ) is the biordered set of the monoid of all 3×3 matrices over F , then the free idempotent generated semigroup over E3(F ) has a subgroup isomorphic to the multiplicative group of F . In particular, letting 2000 Mathematics Subject Classification. 20M05. The second author was supported in part by the NSF grant DMS-0700811 and by a BSF (USA-Israeli) grant. The third author acknowledges support from the Russian Foundation for Basic Research, grant 06-01-00613.