Some nilpotent and commutative finite semigroups Manuel Delgado WORKSHOP ON SEMIGROUPS (To remember John M. Howie on the occasion of his 76th birthday) Lisbon, 23-25 May, 2012
Numerical Semigroups Quotients Computations Nilpotent semigroups Outline Numerical Semigroups 1 Quotients 2 Computations 3 Nilpotent semigroups 4 M. Delgado Some finite semigroups Lisbon, 24/05/2012 2 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups Numerical semigroups Notation N – non-negative integers A semigroup ( N , +) – non-negative integers under the usual addition Subsemigroups Let { 0 } � = S ≤ N and let d = gcd ( S ). � s S � d = d ∈ S | s ∈ S is isomorphic to S . � N \ S � ♯ < ∞ d A reference: J. Rosales and P. Garc´ ıa S´ anchez. Numerical Semigroups . Springer, 2009. M. Delgado Some finite semigroups Lisbon, 24/05/2012 3 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups Definition of numerical semigroup S ⊆ N \ { 0 } S + S ⊆ S ♯ ( N \ S ) < ∞ (equivalently gcd ( S ) = 1) For historical reasons (?) 0 ∈ S is usually required. Every subsemigroup of ( N , +) is isomorphic to a numerical semigroup. Remark Numerical semigroups are isomorphic if and only if they are equal. Remark A numerical semigroup S has unique minimal generating system: S \ ( S + S ). M. Delgado Some finite semigroups Lisbon, 24/05/2012 4 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups Example Consider S = � 3 , 7 , 11 � . Its elements are the amounts that can be formed with the following coins: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 M. Delgado, J. Morais, and P. Garc´ ıa-S´ anchez. Numericalsgps – a GAP package for computing with numerical semigroups, 2011, December. Version number 0.971. http://cmup.fc.up.pt/cmup/mdelgado/numericalsgps/ M. Delgado Some finite semigroups Lisbon, 24/05/2012 5 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups Notable elements Example S = � 5 , 8 , 11 , 14 , 17 � = { a 5+ b 8+ c 11+ d 14+ e 17 | a , b , c , d , e ∈ N , abcde � = 0 } 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 F ( S ) c ( S ) m ( S ) H ( S ) = N \ S = { 1 , 2 , 3 , 4 , 6 , 7 , 9 , 12 } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gaps g ( S ) = ♯ H ( S ) = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . genus m ( S ) = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .multiplicity F ( S ) = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frobenius number c ( S ) = 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .conductor Ongoing experiments led to the following figure (under construction) which exhibits an impressive regularity: M. Delgado Some finite semigroups Lisbon, 24/05/2012 6 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups GENUS 100,000,000,000,000,000,000 2 to 100 1,000,000,000,000,000,000 10,000,000,000,000,000 100,000,000,000,000 1,000,000,000,000 Number of Numerical semigroups 10,000,000,000 100,000,000 1,000,000 10,000 100 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 Multiplicities M. Delgado Some finite semigroups Lisbon, 24/05/2012 7 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups Each column corresponds to a multiplicity and shows the accumulated number of numerical semigroups having that multiplicity up to genus 100. The y-axis is in logarithmic scale. The leftmost part reflects the computations done while the upper rightmost part follows from a result of Kaplan. Number of numerical semigroups of genus 55 1 , 142 , 140 , 736 , 859 We plan also to produce a library of numerical semigroups of small genus. Besides all numerical semigroups up to a certain genus (this number will depend of the space, which in turn depends on a compromise between the compression and the time needed to access the data), the library should contain all the “interesting” numerical semigroups up to a higher genus. M. Delgado Some finite semigroups Lisbon, 24/05/2012 8 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups Rees Quotients Ideal A (semigroup) ideal I of a numerical semigroup S has a unique finite minimal system of generators G such that I = G + ( S ∪ { 0 } ). Example S = � 5 , 7 , 11 � I = { 10 } + ( S ∪ { 0 } ) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Rees quotient S / I = { 5 , 7 , 11 , 12 , 14 , 16 , 18 , 19 , 23 , ∞} M. Delgado Some finite semigroups Lisbon, 24/05/2012 9 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups The quotients obtained this way are commutative nilpotent semigroups. Denote by RQNS the class of semigroups that are Rees quotients of numerical semigroups. We consider also the classes N of nilpotent (finite) semigroups, Com of commutative (finite) semigroups. With V.H. Fernandes we obtained the following: Theorem 2.1 The class RQNS generates the pseudovariety N ∩ Com . Remark A minimal set of generators of S / I is given by the generators of S that are “finite” (i.e., do not belong to I ). M. Delgado and V.H. Fernandes. Quotients of numerical semigroups . Ongoing... M. Delgado Some finite semigroups Lisbon, 24/05/2012 10 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups 5 7 11 Cayley graph 5 12 Computations 19 compute an isomorphic transformation 7 14 semigroup; compute the Cayley 23 graph and use some of the many options 16 available in the GAP package “viz” to display it. 11 18 M. Delgado Some finite semigroups Lisbon, 24/05/2012 11 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups How to compute with these semigroups? The most reasonable approach seems to be to compute an isomorphic transformation semigroup and then use “citrus”. The transformation semigroup can be computed in a somehow clever way: to each generator associate a transformation of the set of elements of the semigroup with an added identity that describes the action of the generator in the semigroup elements. Then, consider the semigroup generated by the transformations obtained. Doing so, our transformation semigroup is given by the minimum number of generators. Note: Constructing the semigroup by the multiplication table and using the right regular representation to obtain a transformation semigroup to do further computations using “Citrus” was too slow. In particular, getting a “small” generating set is slow... M. Delgado Some finite semigroups Lisbon, 24/05/2012 12 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups Code has been written to compute the isomorphic transformation semigroup. Options have been added to “viz” to allow to have labellings as the ones in the Cayley graph shown... Software: M. Delgado, A. Egri-Nagy, J. D. Mitchell and M. Pfeiffer. Viz - a GAP package for drawing GAP objects. Under development. https://bitbucket.org/zen154115/viz J. D. Mitchell. Citrus - a GAP package to compute with transformation semigroups, 2012. Version number 0.6. http://www-groups.mcs.st-andrews.ac.uk/~jamesm/citrus/ The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.5 (beta), 2012. http://www.gap-system.org/ M. Delgado Some finite semigroups Lisbon, 24/05/2012 13 / 14
Numerical Semigroups Quotients Computations Nilpotent semigroups The motivation for the computational work referred comes from work (some already done, other ongoing) of A. Distler concerning the classification of nilpotent semigroups. Having examples at hand can be useful to get intuition that can help to proceed. A. Distler. Classification and enumeration of finite semigroups . PhD thesis, University of St Andrews, 2010. http://research-repository.st-andrews.ac.uk/handle/10023/945 M. Delgado Some finite semigroups Lisbon, 24/05/2012 14 / 14
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