FAITHFUL AND TRANSITIVE MONOID ACTIONS AMS - EMS - SPM International Meeting Porto, 2015 Enric Cosme-Llópez Departament d'Àlgebra Universitat de València • Generalisation of group actions • Automata Theory Introduction Related Work Our Work Bibliography ACTIONS An action of a monoid M on a nonempty set Ω is a monoid homomorphism : M −! TΩ = ff j f :Ω ! Ωg 2 Introduction Related Work Our Work Bibliography ACTIONS An action of a monoid M on a nonempty set Ω is a monoid homomorphism : M −! TΩ = ff j f :Ω ! Ωg • Generalisation of group actions • Automata Theory 2 Objective Give a description of transitive transformation monoids Introduction Related Work Our Work Bibliography ACTIONS The action is faithful if the corresponding monoid homomorphism is injective. We refer to the pair (M; Ω) as transformation monoid. The action is transitive if for each α; β 2 Ω, the cyclic sets coincide αM = βM. 3 Introduction Related Work Our Work Bibliography ACTIONS The action is faithful if the corresponding monoid homomorphism is injective. We refer to the pair (M; Ω) as transformation monoid. The action is transitive if for each α; β 2 Ω, the cyclic sets coincide αM = βM. Objective Give a description of transitive transformation monoids 3 It is based on Module theory - Green Semigroup representation - Clifford, Munn, Ponizovsky Introduction Related Work Our Work Bibliography STEINBERG Related Work [Ste10] B. Steinberg. A Theory of Transformation Monoids: Combinatorics and Representation Theory. Electron. J. Combin., 17(1): Research Paper 164, 56 pp., 2010. 4 Introduction Related Work Our Work Bibliography STEINBERG Related Work [Ste10] B. Steinberg. A Theory of Transformation Monoids: Combinatorics and Representation Theory. Electron. J. Combin., 17(1): Research Paper 164, 56 pp., 2010. It is based on Module theory - Green Semigroup representation - Clifford, Munn, Ponizovsky 4 ] ] ] eM = Gej = Hxiej ej2E(eM) xi2T ej2E(eM) Introduction Related Work Our Work Bibliography PRELIMINARIES Proposition Any transitive and faithful action of a monoid M on a set Ω is equivalent to the action of M on eM/≡ where e is an idempo- tent in I(M) and ≡ is a right-congruence on eM. 5 Introduction Related Work Our Work Bibliography PRELIMINARIES Proposition Any transitive and faithful action of a monoid M on a set Ω is equivalent to the action of M on eM/≡ where e is an idempo- tent in I(M) and ≡ is a right-congruence on eM. ] ] ] eM = Gej = Hxiej ej2E(eM) xi2T ej2E(eM) 5 For each element xi in a right-transversal T, we define an equivalence relation 'i in the set n = f1; ··· ; ng as follows i 0 0 j ' j , Hxi(j)ρm = Hxi(j )ρm; for all m 2 M Introduction Related Work Our Work Bibliography PRELIMINARIES M embeds in Ge on M via the map m 7! (ρm; m), where (j)ρm = g such that ejm = gej0 6 Introduction Related Work Our Work Bibliography PRELIMINARIES M embeds in Ge on M via the map m 7! (ρm; m), where (j)ρm = g such that ejm = gej0 For each element xi in a right-transversal T, we define an equivalence relation 'i in the set n = f1; ··· ; ng as follows i 0 0 j ' j , Hxi(j)ρm = Hxi(j )ρm; for all m 2 M 6 Introduction Related Work Our Work Bibliography ON TRANSITIVE TRANSFORMATION MONOIDS Theorem If (M; Ω) is a transitive transformation monoid, there exists a core-free subgroup H of Ge such that every member of the partition fHxiej j xi 2 T; ej 2 E(eM)g is contained in a ≡-class. Furthermore, 0 i 0 i) If Hxiej≡Hxi0 ej0 then i = i and j ' j ; ii) If Hxiej≡Hxiej0 then (Hxiej)m≡(Hxiej0 )m for all m 2 M; iii) For different idempotents ej; ej0 there exists some xi 2 T with Hxiej and Hxiej0 not ≡-related. The converse of this Theorem also holds. 7 Introduction Related Work Our Work Bibliography ON TRANSITIVE TRANSFORMATION MONOIDS Consequences • We can construct all transitive and faithful actions of a monoid that restrict to a given transitive and faithful group action. • There exist non-equivalent transitive and faithful monoid actions that restrict to the same group action. • These theorems extend the classical theorems on the description of transitive permutation groups. 8 A Theory of Transformations Monoids [Ste10] ''We hope that the theory of primitive permutation groups can be used to understand primitive transformation monoids in the case the maximal subgroups of I(M) are non-trivial'' Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS An action of a monoid M on a set Ω is said to be primitive if it admits no non-trivial proper congruences. 9 Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS An action of a monoid M on a set Ω is said to be primitive if it admits no non-trivial proper congruences. A Theory of Transformations Monoids [Ste10] ''We hope that the theory of primitive permutation groups can be used to understand primitive transformation monoids in the case the maximal subgroups of I(M) are non-trivial'' 9 Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS Theorem Let M be a transitive transformation monoid such that Ge is non-trivial. Then M is primitive if and only if there exists a maximal core-free subgroup H of Ge such that the relations ' defined by H in E(eM) satisfy the following conditions i) For different idempotents ej; ej0 there exist finite s s+1 sequences (xit )t=1 and (ejt )t=1 of elements of T and E(eM), respectively, such that i1 i2 is−1 is 0 j = j1 ' j2 ' ···' js ' js+1 = j ii) For different idempotents ej; ej0 there exists some xi 2 T with j 6'i j0. 10 Introduction Related Work Our Work Bibliography ON PRIMITIVE TRANSFORMATION MONOIDS Let (M; Ω) be a transitive transformation monoid. Consequences • If Ge has a primitive action, there exists a unique primitive action of M that restricts to the primitive action of Ge. • If Ge is a non-trivial nilpotent group. Then M has a primitive action if and only if M is a cyclic group of prime order. • If jΩej = 2, then the action of M on Ω cannot be primitive. 11 Introduction Related Work Our Work Bibliography BIBLIOGRAPHY A. Ballester-Bolinches, E. Cosme-Llópez, P. Jiménez-Seral, Some Contributions to the Theory of Transformation Monoids, Preprint, 2015. J. Rhodes and B. Steinberg, The q-theory of finite semigroups, Springer Monographs in Mathematics. Springer, 2009. B. Steinberg, A theory of transformation monoids: Combinatorics and representation theory, The Electronic Journal of Combinatorics, 17:1–56, 2010. HSRM Beamer Theme, by Benjamin Weiss. 12.