COMPLEXITY PSEUDOVARIETIES ARE NOT LOCAL; TYPE II SUBSEMIGROUPS CAN FALL ARBITRARILY IN COMPLEXITY JOHN RHODES AND BENJAMIN STEINBERG Abstract. We prove the following two results announced by Rhodes: The Type II subsemigroup of a finite semigroup can fall arbitrarily in complexity; the complexity pseudovarieties Cn (n ≥ 1) are not local. 1. Introduction and Key Results It is asserted in [5] that there are semigroups (all semigroups in this paper are finite) of arbitrary complexity in (A m  G) m  G (Rhodes unpublished, 1972). Equivalently, it is asserted that if S is a semigroup, then the Type II subsemigroup [5, 15] K G (S ) of S can drop arbitrarily in complexity. In [11] Rhodes constructs examples of semigroups of complexity two and three in (A m  G) m  G. The unpublished examples of Rhodes were obtained via Kernel Systems [11], iterated matrix semigroup theory [23, 12, 13] and the Presentation Lemma [2]; the proofs were too cumbersome to write up. Similarly it is asserted [22] that the complexity pseudovarieties C n (n> 1) are not local in the sense of Tilson (Rhodes unpublished). This follows for complexity one and complexity two by the results of [11], but again the general case was simply too messy to publish. Steinberg has since devel- oped a simplification of the Presentation Lemma [19] and some new ideas on working with iterative matrix semigroups [14] (used to prove the com- plexity pseudovarieties are not finitely based) that allow for a reasonably clean presentation of these results of Rhodes. Let A be the pseudovariety of aperiodic semigroups and G the pseudova- riety of groups. Define C 0 = A and, inductively, C n = A ∗ G ∗ C n−1 , where ∗ denotes the semidirect product of pseudovarieties [3]. The Krohn- Rhodes theorem [7] says that the C n exhaust the pseudovariety of all semi- groups. The complexity [8] of a semigroup S , denoted c(S ), is then the least n for which S ∈ C n . Date : May 6, 2004. 1991 Mathematics Subject Classification. 20M07. Key words and phrases. Complexity, Presentation Lemma, Type II. The second author was supported in part by NSERC and by POCTI approved project POCTI/32817/MAT/2000 in participation with the European Community Fund FEDER. 1