On the structure of acts over semigroups

Transcription

1 On the structure of acts over semigroups A.V. (Moscow) A.V. Moscow State University, National Research University MIET 1/ 16

2 Happy Birthday to Ulrich Knauer! A.V. Moscow State University, National Research University MIET 2/ 16

3 Definitions S semigroup, X a right S-act: X S X, (x, s) xs s.t. x(ss ) = (xs)s for x X, s, s S. Left S-act: (ss )x = s(s x). (S, T )-bi-act: (sx)t = s(xt) for x X, s S, t T. Multi-act: (xs i )s j = (xs j )s i for s i S i, s j S j, i j. Partial act: x(ss ) = (xs)s iff the left side exists Act = automaton = unary algebra A.V. Moscow State University, National Research University MIET 3/ 16

4 Publications Glushkov V.M. The abstract theory of automata. Uspekhi Mat. Nauk, 1961 Plotkin B.I., Gringlaz L.Ya., Gvaramiya A.A. Elements of the algebraic theory of automata, Moscow, Arbib M., Krohn K., Rhodes J. Algebraic theory of machines, languages, and semi-groups. Acad. Press, Eilenberg S. Automata, languages and machines/ Ac. Press, 1976 Lallement G. Semigroups and combinatorial applications. PA State Univ., Kilp M., Knauer U., Mikhalev A.V. Acts, monoids and categories. De Gruyter, A.V. Moscow State University, National Research University MIET 4/ 16

5 Acts over completely (0-)simple semigroups G group L left zero semigroup R right zero semigroup L R rectangular band L G R rectangular group M(G, I, Λ, P) comletely simple semigroup M 0 (G, I, Λ, P) compl. 0-simple sgp Theorem (Avdeyev, Kozhukhov, 2000). Let X be a set, Q = γ Γ (G/H γ), κ λ : Q X (λ Λ) and π i : X Q (i I ) be mappings such that qκ λ π i = q p λi. Put x (g) iλ = (xπ i g)κ λ. Then X is an M-act. Moreover, any M-act is isomorphic to an act constructed by this way. Similar statement takes place for the acts with zero over M 0. Haliullina A.R : a description of congruences of acts over G, L, R A.V. Moscow State University, National Research University MIET 5/ 16

6 Acts over semilattices Semilattice = p.o.set with inf = commutative idempotent sgp An act (even partial act) X over a semilattice S is a p.o.set: x y x ys 1. Apraksina T.V., Maksimovskiy M.Yu. ( ): Some necessary conditions and some sufficient conditions are found for extension operation partial complete. A sufficient condition is the minimal condition. Kozhukhov I.B., Maksimovskiy M.Yu.(2010): a complete description of acts over a finite chain A.V. Moscow State University, National Research University MIET 6/ 16

7 Injectivity and projectivity Kilp M., Knauer U., Mikhalev A.V., Chap. III, IV Berthiaume P. (1967) the injective envelope of act Isbell J.R. (1971) the projective cover (the conditions of existence) Moggaddasi Gh. (2012): the conditions of injectivity, the injective envelope of acts over L by assumption of separability: x y s 1 xs ys Kozhukhov I.B., Petrikov A.O. ( ): conditions of injectivity, projectivity, constructions of injective envelope, projective cover over M, M 0 without assumptions S = M(G, I, Λ, P); S S = i I R i where R i = (e) iλ S. X is projective X is isomorphic to a coproduct of copies of (S 1 ) S and R i (R i = Rj for i, j I ) Earlier: Haliullina A.R., Kozhukhov I.B. - over L, R, L R A.V. Moscow State University, National Research University MIET 7/ 16

8 Congruences of acts Kartashova A.V. (2011) On the lattices of congruences and topologies of unary algebras. The connection of the lattices of congruences, topologies and quasi-orders were investigated. It is established that QOrdA is embedded into TopA and they are coincided with one another in case when A <. The conditions of modularity, distributivity, representability are found. Radeleczki S. (1996): the connections of ConA and the group AutA (A is a unary algebra) Stepanova A.A., Ptakhov D.O. (2013): the conditions of modularity, distributivity, and "to be a chain"for ConX where X is a non-connected act (as a graph) Haliullina A.R. ( ): a description of acts X over S such that ConX is modular, or distributive, or a chain, for S = R, S = L. For these acts: X 9, ConX 160, S 27 (if the action is effective) A.V. Moscow State University, National Research University MIET 8/ 16

9 Subdirectly irreducible acts ConX has: = {(x, x) x X } (the least congruence), = {(x, y) x, y X } = X X (the greatest congr.) Birkhoff G.: any algebra is a subdirect product of subdirectly irreducible algebras; ConX has the least non-trivial (i.e. ) congruences Yoeli M. (1967): subdirectly irreducible unars are described Ésik Z., Imreh B. (1981): subd. irr. commutative automata are described Roiz E.N. (1974): subd. irr. acts over arbitrary sgps are investigated Kozhukhov I.B., Haliullina A.R.(2015): subd. irr. acts are characterized "up to the core"(the least subact) subd. irr. acts over L R are described A.V. Moscow State University, National Research University MIET 9/ 16

10 Semigroups with finitely approximated acts Consider 2 classes of semigroups: ( ) semigroups whose acts are finitely approximated ( ) semigroups whose acts are approximated by acts of n elements Kozhukhov I.B. (1999): ( ) S/ρ is fin. appr. (ρ is a right congruence) Kozhuhov I.B., Haliullina A.R. (2015): 1) a group satisfies ( ) any subgroup is an intersection of subgroups of finite indices, 2) if S S is uniformly fin. approx. then S is uniformly locally finite Kozhukhov I.B., Tsarev A.V. (2015): a complete description of abelian groups satisfying ( ) or ( ): A satisfies ( ) A is bounded (ma = 0 for some m) A.V. Moscow State University, National Research University MIET 10/ 16

11 Diagonal acts (S S) S right diagonal act, S (S S) left diagonal act S(S S) S diagonal bi-act (S n ) S, S (S n ), S (S n ) S diagonal act of n-th order rdrs, ldrs, bdrs, rdr n S,... diagonal ranks: rdrs = min{ A : A S S & S 1 AS 1 = S S} When rdrs <? rdrs = 1? The same for ldrs, bdrs. Gallagher P., Ru skuc N. (2005): rdrs =ldrs = 1 if S = T X, or P X, or B X where X is an infinite set Gallagher (2006): conditions for the act (or bi-act) to be cyclic, finitely generated A.V. Moscow State University, National Research University MIET 11/ 16

12 Connections between diagonal acts of different orders Theorem 1 (Apraksina T.V., Barkov I.V., Kozhukhov I.B., 2015). If S is infinite and rdrs < then rdr n S (rdrs) n For bi-acts: Theorem 2 (Apr., Bark., Kozh., 2013). Any semigroup S can be embedded into a semigroup T such that the diagonal bi-act T (T T ) T is cyclic but the diagonal bi-act T (T T T ) T is not finitely generated. A.V. Moscow State University, National Research University MIET 12/ 16

13 Finite generated and cyclic diagonal acts Theorem (Apr., Bark., Kozh., 2015). If S is infinite semigroup satisfying a non-trivial law then (S S) S is not fin. gen. and S (S S) S is not cyclic. Gallagher, 2006: If S is an infinite inverse sgp. then (S S) S is not fin. gen. {finite} {loc. fin.} {periodic } {epi-groups} Theorem (Kozhukhov I.B., Olshanski A. Yu., 2015). S is loc. fin. & bdrs < S <. A.V. Moscow State University, National Research University MIET 13/ 16

14 Table of results Semigroup S S > 1 rdrs = 1 S = rdrs < S > 1 bdrs = 1 S = bdrs < Commutative No No No No Idempotent No No No No With identity No* No*??? Yes* Inverse No No Yes Yes Completely regular No No No No Completely 0-simple No No No Yes Group No No No Yes Cancellative No No No Yes Right cancellative No No Yes* Yes Left cancellative No No Yes* Yes Epigroup No* No* No* Yes Periodic No* No* No* Yes Locally finite No No No* No* Right invariant No* No* Yes* Yes* Left invariant No* No* Yes* Yes* A.V. Moscow State University, National Research University MIET 14/ 16

15 Diagonal ranks of completely (0-)simple semigroups Barkov I.V., Shakirov R.R. (2015) Theorem 1. If S = M(G.I.Λ, P), G = t, I = k, Λ = l then квк;ы=л:2е; for l = 1, = k 2 t 2 l(l 1) for l > 1. Theorem 2. Let S = M 0 (G.I.Λ, P). for l 2: rdrs = k 2 t 2 (l 2 l) + 2k if P does not contain zeros, rdrs = k 2 t 2 (l 2 l) + k 2 t if P has a zero but there are no column with 2 non-zero elements, rdrs = k 2 t 2 (l 2 l) otherwise, for l = 1 rdrs = k 2 t + 2k. Barkov I.V.: bdrm, bdrm 0. A.V. Moscow State University, National Research University MIET 15/ 16

16 Semigroups of minimal diagonal rank Let S = n. Then n rdrs n 2. Theorem (Barkov I.V. 2014). Let S be a semigroup of n elements. rdrs = n iff one of conditions holds: 1 S is a group; 2 S left zero sgp of 2 elements; 3 S sgp defined by table: e a b e e a b a b b b b b b b A.V. Moscow State University, National Research University MIET 16/ 16

17 Diagonal acts over semigroups of continuous transformations Isotone transformation may be considered as continuous mappings (the topology is defined by the order) Theorem (Apraksina T.V., 2011). Let X be a p.o.set. The act (O(X ) O(X )) O(X ) is cyclic there exist subsets X 1, X 2 X such that X 1 = X2 = X and x y, y x for x X1, y X 2. Other results: 1 If X = [0, 1] then the right diagonal act (C(X ) C(X )) C(X ) is cyclic but the left diagonal act C(X ) (C(X ) C(X )) has 2 ℵ0 generators 2 Conditions for being cyclic diagonal acts over semigroup of partial isotone transformations 3 bdro(z) > ℵ 0 A.V. Moscow State University, National Research University MIET 17/ 16

18 A.V. Moscow State University, National Research University MIET 17/ 16