Ranks of Semigroups Nik Ruškuc nik@mcs.st-and.ac.uk School of Mathematics and Statistics, Lisbon, 24 May 2012
Prologue... the word mathematical stems from the Greek expression ta mathemata, which means what can be learned and thus, at the same time, what can be taught;... Teaching therefore does not mean anything else then to let others learn, that is, to bring one another to learning. (M. Heidegger)
John Howie on Generators and Ranks J.M. Howie, The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc. 41 (1966), 707 716. J.M. Howie, Products of idempotents in certain semigroups of transformations, Proc. Edinburgh Math. Soc. (2) 17 (1970/71), 223 236. J.M. Howie, Some subsemigroups of infinite full transformation semigroups Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159 167. E. Giraldes, J.M. Howie, Semigroups of high rank, Proc. Edinburgh Math. Soc. 28 (1985), 13 34. J.M. Howie, R.B. McFadden, Idempotent rank in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 161 167. G.M.S. Gomes, J.M. Howie, On the ranks of certain semigroups of order preserving transformations, Semigroup Forum 45 (1992), 272 282. J.M. Howie, P.M. Higgins, N. Ruškuc, On relative ranks of full transformation semigroups, Comm. Algebra 26 (1998), 733 748. J.M. Howie,M.I.M. Ribeiro, Rank properties in finite semigroups, Comm. Algebra 27 (1999), 5333 5347. P.M. Higgins, J.M. Howie, J.D. Mitchell, N. Ruškuc, Countable versus uncountable ranks in infinite semigroups of transformations and relations, Proc. Edin. Math. Soc. 46 (2003), 531 544. G. Ayık, H. Ayık, Y. Ünlu, J.M. Howie, Rank properties of the semigroup of singular transformations on a finite set, Comm. Algebra 36 (2008), 2581 2587.
Products of idempotents in T X : finite X J.M. Howie, The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc. 41 (1966), 707 716. T X the full transformation semigroup. E X = E(T X ) the idempotents of T X. If X = {1,..., n} then E X = Sing X = {α : im(α) < n}.
Products of idempotents in T X : infinite X S(α) = {x X : xα x}; Z(α) = X \ X α; C(α) = {xα 1 : xα 1 2}; s(α) = S(α) shift. z(α) = Z(α) defect. c(α) = C(α) collapse. If X is infinite then E X = {α : s(α) <, z(α) > 0} {α : s(α) = z(α) = c(α) ℵ 0 }.
Vista Clear finite/infinite separation; numbers of generators; lengths of products; other semigroups (e.g. K(n, r), O X ).
Rank: full transformations rank(s) = min{ A : A = S}. idrank(s) = min{ A : A E(S), A = S}. (Gomes, Howie 87) rank(sing n ) = idrank(sing n ) = n(n 1)/2. K(n, r) = {α T n : imα r}. (McFadden, Howie 90) rank(k(n, r)) = idrank(k(n, r)) = S(n, r), the Stirling number. Remark S(n, r) = number of R-classes in the D-class {α : im(α) = r}.
Rank: order preserving transformations G.M.S. Gomes, J.M. Howie, On the ranks of certain semigroups of order preserving transformations, Smgp For. 45 (1992), 272 282. O n = order preserving transformations of {1,..., n}. rank(o n ) = n, idrank(o n ) = 2n 2. Remark The top J -class of O n has n 1 R-classes and n L-classes. PO n = partial order preserving transformations of {1,..., n}. rank(po n ) = 2n 1, idrank(o n ) = 3n 2. Remark The top J -class of PO n has 2n 1 R-classes and n L-classes.
Vista Often rank(s) = rank(p), where P is the principal factor corresponding to a unique maximal J -class. Why? Often rank(p) = max( P/L, P/R ). Why? Sometimes rank(s) = idrank(s). Why? How to find the rank of a general completely 0-simple semigroup?
Completely 0-simple semigroups: connected case S = M 0 [G; I, Λ; P] a Rees matrix semigroup. Bipartite graph Γ(P): V = I Λ; i λ p λi 0. H = a subgroup of G depending on P and Γ. H {p λi : λ Λ, i I } \ {0} :-) Relative rank: rank(g : H) = min{a : H A = G}. (NR 94) If Γ is connected then Corollary If S is completely simple then rank(s) = max( I, Λ, rank(g : H)). rank(s) = max( I, Λ, rank(g : H)), where H = p λi : λ Λ, i I.
Completely 0-simple semigroups: general case S = M 0 [G; I, Λ; P] a Rees matrix semigroup. Connected components: I 1 Λ 1,..., I k Λ k. H i = subgroups of G depending on P and Γ (i = 1,..., k). (Gray, NR 05) rank(s) = max( I, Λ, r + k 1), where r = min{rank(g : k i=1 g 1 i H i g i ) : g i G}.
Vista Gray 08: A combinatorial condition (Hall-like) for rank(s) = idrank(s) (S completely simple). Ideas of connectedness. Recent work on free idempotent generated semigroups Gray, Dolinka, NR. Semigroups of high rank (Giraldes, Howie 85). Different notions of rank (Ribeiro, Howie 99, 00).
Depth in infinite semigroups J.M. Howie, Some subsemigroups of infinite full transformation semigroups, Proc.Roy.Soc.Edin. Sect. A 88 (1981), 159 167. Depth of S w.r.t. generating set A = smallest n such that S = n i=1 An. Recall: E X = F X ( ℵ 0 m X Q m), where F = {α : s(α) <, d(α) > 0}, Q m = {α : s(α) = z(α) = c(α) = m}. (i) Depth( E X ) = Depth(F ) = ; (ii) Depth(Q m ) = Depth( E X \ F ) = 4.
Relative ranks in infinite semigroups P.M. Higgins, J.M. Howie, N. Ruškuc, On relative ranks of full transformation semigroups, Comm.Alg. 26 (1998), 733 748. rank(t X : S X ) = 2. K(α) = {x X : xα 1 = X }; k(α) = K(α) infinite contraction index. S X, µ, ν = T X iff µ is an injection with d(µ) = X and ν is a surjection with k(ν) = X (or v.v.). rank(t X : E(T X ) ) = 2.
Relative ranks in infinite semigroups P.M. Higgins, J.M. Howie, J.D. Mitchell, N. Ruškuc, Countable versus uncountable ranks in infinite semigroups of transformations and relations, Proc.Edin.Math.Soc. 46 (2003), 531 544. (Sierpiński 35) For any τ 1, τ 2, T X there exist µ, ν T X such that τ i µ, ν. Corollary For every S T X either rank(t X : S) 2 or else rank(t X : S) is uncountable. Analogous results hold for: S X (Galvin 95), B X, I X. rank(b X : T X ) = rank(b X : I X ) = rank(t N : O N ) = 1. rank(b X : S X ) = rank(b X : Surj X ) = rank(b X : Inj X ) = 2. Further work: J.D. Mitchell, M. Morayne, Y. Peresse, J. Cichon, M. Quick, P. Higgins, NR.
Sierpinski rank Definition The Sierpinski rank of S is the smallest number n such that for any s 1, s 2, S there exist a 1,..., a n S such that s i a 1,..., a j. Proposition srank(s X ) = srank(t X ) = srank(i X ) = srank(p X ) = srank(b X ) = 2. (Peresse 09) { n + 4 if X = ℵn srank(inj X ) = otherwise. (Mitchell, Peresse 11) srank(surj X ) = { n 2 +9n+14 2 if X = ℵ n otherwise.
Bergman property V. Maltcev, J.D. Mitchell, N. Ruškuc, The Bergman property for semigroups, J. Lond. Math. Soc. 80 (2009), 212 232. Definition S has Bergman property if it has finite depth w.r.t. every generating set. (Bergman 06) S X has Bergman property. T X, P X, I X, B X all have Bergman property. The finitary power semigroup of I X has Bergman property, but those of T X, P X, B X do not.
d-sequences and relative rank NR, Quick, Geddes, Hyde, Wallis, Loughlin, Carey, Awang, McLeman, Garrido. d i = rank(s i ) (direct power). d(s) = (d 1, d 2, d 3,... ). Relate asymptotic properties of d(s) and algebraic properties of S. n (S) = {(s,..., s) : s S} S n. d i = rank(s n : n (S)). d(s) = (d 1, d 2, d 3,... ). Proposition d(s) d(s).
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