For any group $G$ and integer $k\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $AC_k(G)$, on the subset $N_k(G) \subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup (provided $N_k(G)$ is non-empty). The famous Andrews-Curtis Conjecture is that if $G$ is free of rank $k$, then $AC_k(G)$ acts transitively on $N_k(G)$. The set $N_k(G)$ may have a rather complex structure, so it is easier to study the full Andrews-Curtis group $FAC(G)$ generated by AC-transformations on a much simpler set $G^k$. Our goal here is to investigate the natural epimorphism $λ\colon FAC_k(G) \to AC_k(G)$. We show that if $G$ is non-elementary torsion-free hyperbolic, then $FAC_k(G)$ acts faithfully on every nontrivial orbit of $G^k$, hence $λ\colon FAC_k(G) \to AC_k(G)$ is an isomorphism.7 pages. In memory of Ben Fine. Published in journal of Groups, Complexity, Cryptology

In this paper we, contribute the notation of natural epimorphism of a semilattice on the quotient semilattice and subsemilattice. If S is distributive semilattice and F is a filter of S, then we demonstrate that θF is the smallest congruence on S containing F in a single equivalence class and that S/θF is distributive. In addition, the author proved that map FθF is an isomorphism from the lattice of F0(S) all non-empty filters of S into a permutable sublattice of the lattice C(S) of all congruencies on S.

Let be a regular odd prime, the th cyclotomic field and , where is a positive integer. Under the assumption that there are exactly three places not over that ramify in , we continue to study the structure of the Tate module (Iwasawa module) as a Galois module. In the case , we prove that for finite we have for some odd positive integer . Under the same assumptions, if is the Galois group of the maximal unramified Abelian -extension of , then the kernel of the natural epimorphism is of order . Some other results are obtained.

Congruence topologies on the mapping class group

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any (possibly infinite) tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text] (so a tuple of unbounded length). By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see text] are naturally [Formula: see text]-flows (even [Formula: see text]-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of [Formula: see text]), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called [Formula: see text]-topology) on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of [Formula: see text]) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 (2018) 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism (in particular, [Formula: see text] is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow [Formula: see text] is replaced by [Formula: see text].

Amenability, definable groups, and automorphism groups

Abstract For a groupGdefinable in a first order structureMwe develop basic topological dynamics in the category of definableG-flows. In particular, we give a description of the universal definableG-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification ofG, that is to the quotient${G^{\rm{*}}}/G_M^{{\rm{*}}00}$(whereG* is the interpretation ofGin a monster model). More generally, we obtain these results locally, i.e., in the category of Δ-definableG-flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components$G_{{\rm{\Delta }},M}^{{\rm{*}}00}$and$G_{{\rm{\Delta }},M}^{{\rm{*}}000}$, and show that${G^{\rm{*}}}/G_{{\rm{\Delta }},M}^{{\rm{*}}00}$is the Δ-definable Bohr compactification ofG. We also note that some deeper arguments from [14] can be adapted to our context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification ofG. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

The concept of an I-matrix in the full 2 × 2 matrix ring M2(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix [Formula: see text] in M2(R/I) as the sum of two subrings 𝒮1 and 𝒮2 of M2(R/I), where 𝒮1 is the image (under the natural epimorphism from M2(R) to M2(R/I)) of the centralizer in M2(R) of a pre-image of [Formula: see text], and the entries in 𝒮2 are intersections of certain annihilators of elements arising from the entries of [Formula: see text]. It turns out that if R is a PID, then every matrix in M2(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.

The concept of an I-matrix in the full 2 × 2 matrix ring M 2(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, was introduced in a Marais article that is to be published [10]. Moreover, a concrete description of the centralizer of an I-matrix in M 2(R/I) as the sum of two subrings 𝒮1 and 𝒮2 of M 2(R/I) was also given, where 𝒮1 is the image (under the natural epimorphism from M 2(R) to M 2(R/I)) of the centralizer in M 2(R) of a pre-image of , and where the entries in 𝒮2 are intersections of certain annihilators of elements arising from the entries of . In the present paper, we obtain results for the case when I is a principal ideal⟨k⟩, k ∈ R a nonzero nonunit. Mainly we solve two problems. Firstly we find necessary and sufficient conditions for when 𝒮1 ⊆ 𝒮2, for when 𝒮2 ⊆ 𝒮1 and for when 𝒮1 = 𝒮2. Secondly we provide a formula for the number of elements in the centralizer of for the case when R/⟨k⟩is finite.

Here we study the structure of a group with the presentation by conjugation which has been assigned to an extended affine root system R. We show that is isomorphic to the direct product of the extended affine Weyl group 𝒲 of R and the kernel of a natural epimorphism . The article also shows that has structural features that are well known for 𝒲.

Let F be a free pro-p group of finite rank n and \({C_{p^r}}\) a cyclic group of order pr. In this work we classify p-adic representations \({ C_{p^r}\longrightarrow GL_n(\mathbb{Z}_{p})}\) that can be obtained as a composite of an embedding \({C_{p^r}\longrightarrow {\rm Aut}(F)}\) with the natural epimorphism \({{\rm Aut}(F)\longrightarrow GL_n(\mathbb{Z}_{p})}\) .

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, kernel of the multiplication map. We prove in many cases that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra. In our cases, the special fiber rings of the diagonal ideals are the homogeneous coordinate rings of the join varieties.

Let Hv(An) and Hv(Bn) be the Hall algebras over ℚ(v) of the Dynkin quivers An and Bn (n ≥ 1), respectively, where v is an indeterminate and the quivers have linear orientation. By comparing the quantum Serre relations, we find a natural algebra epimorphism π : Hv(Bn) → Hv2(An). We determine the kernel of π by giving two sets of generators. Let φ be the natural algebra homomorphism from Hv(An) to the quantized Schur algebra Sv(n + 1, r)(r ≥ 1) and write [Formula: see text] for the induced map. We obtain several ideals of Hv(Bn) by lifting the kernel of φ to the kernel of the composition map [Formula: see text].

The description of orthogonal groups (and close to them ones) in terms of generatrices and relations represents one of the main problems of the combinatorial theory of linear groups. This paper is dedicated to the mentioned question; namely, here we find generating elements and defining relations of a generalized orthogonal group O◦(n,R), n ≥ 2, over a commutative semilocal ring R (in general, without identity) under certain natural constraints. In order to formulate the problem exactly, let us define several necessary notions. LetΛ be an arbitrary associative ring and let ◦ be its quasi-multiplication (i. e., x ◦ y = x+ xy + y). An element α from Λ is said to be quasi-invertible, if with certain α′ ∈ Λ, α ◦ α′ = 0 = α′ ◦ α. If α is quasi-invertible, then one can uniquely determine its quasi-inverse α′. The set of all quasi-invertible elements Λ◦ from Λ forms a group with respect to the composition ◦ (where the identity is zero). We consider the case when Λ = M(n,R) is a complete matrix ring over a ring T which is (associative-)commutative and not necessarily with identity. Let mean the transposition in Λ. The set of quasi-invertible matrices from Λ such that a′ = a forms a group with respect to the matrix quasi-multiplication. We denote it by O◦(n,R) and call it a generalized orthogonal group of the degree n over the ring R. Note that if R contains the identity, then the mapping O(n,R) → O◦(n,R), E + a → a, where E is the unit matrix of the order n, defines an isomorphism. So the introduced groupO◦(n,R) generalizes the notion of a usual orthogonal group to the most general cases of associative-commutative rings R. Let now R be a commutative semilocal ring (not necessarily with identity) and let J = J(R) be its Jacobson radical. By definition it means that R/J ∼= k1 ⊕ · · · ⊕ km, where ki are certain fields (i = 1, . . . ,m). Let Ri stand for the complete preimage of the addend ki under the natural epimorphism R → R = R/J, x → x = x+ J. (1)

Let T be an integral domain, I a nonzero ideal of T, ϕ: T → T/I the natural ring epimorphism, D an integral domain which is a proper subring of T/I, and R = ϕ−1(D). Let k = qf(D) be the quotient field of D such that k ⊆ T/I, S = ϕ−1(k), and k* = k\\{0}. We prove that if k ⊊ T/I and if the map φ: U(T) → U(T/I)/U(D), given by u ↦ ϕ(u)U(D), is surjective, then Cl(R) = Cl(D) ⊕ Cl(S) and Pic(R) = Pic(D) ⊕ Pic(S). We also prove that the map :U(T) → U(T/I)/k*, given by u ↦ ϕ(u)k*, is surjective if and only if {x ∈ T | ϕ(x) ∈ U(T/I)} = U(T)(S\\I), if and only if the map λ:Cl(S) → Cl(T), defined by [H] ↦ [(HT) t ], is injective.

Let N be the stabilizer of the word w = s 1 t 1 s 1 −1 t 1 −1 … s g t g s g −1 t g −1 in the group of automorphisms Aut(F 2g ) of the free group with generators ⨑ub;s i, t i⫂ub; i=1,…,g . The fundamental group π1(Σg) of a two-dimensional compact orientable closed surface of genus g in generators ⨑ub;s i, t i⫂ub; is determined by the relation w = 1. In the present paper, we find elements S i, T i ∈ N determining the conjugation by the generators s i, t i in Aut(π1(Σg)). Along with an element β ∈ N, realizing the conjugation by w, they generate the kernel of the natural epimorphism of the group N on the mapping class group M g,0 = Aut(π1(Σg))/Inn(π1(Σg)). We find the system of defining relations for this kernel in the generators S 1, …, S g, T 1, …, T g, α. In addition, we have found a subgroup in N isomorphic to the braid group B g on g strings, which, under the abelianizing of the free group F 2g , is mapped onto the subgroup of the Weyl group for Sp(2g, ℤ) consisting of matrices that contain only 0 and 1.

For an anti-congruence q we say that it is regular anti-congruence on semigroup (S,=, _=, ?, ?) ordered under anti-order ? if there exists an antiorder ? on S/q such that the natural epimorphism is a reverse isotone homomorphism of semigroups. Anti-congruence q is regular if there exists a quasi-antiorder ? on S under ? such that q = ? ? ??1. Besides, for regular anti-congruence q on S, a construction of the maximal quasi-antiorder relation under ? with respect to q is shown.

This correspondence presents a multistage decoding approach for a free Zopfq-submodule of Zopfq N of rank K defined by a sparse (N-K)timesN parity-check matrix overZopf q where q=pm, p=2 and m>1. The proposed method involves the repeated application of belief propagation decoding to exploit the natural ring epimorphism ZopfqrarrZopfp l:r|rarr Sigma i=0 l-1r(i)pi with kernel p lZopfq for each l, 1lesllesm, where Sigma i=0 m-1r(i)pi is the p-adic expansion of r. Computer simulations for codes of rate half and moderate length on an additive white Gaussian noise (AWGN) channel with various modulation schemes show that such a decoding strategy offers an additional coding gain of between 0.07-0.1 dB over a single-stage decoding approach

We construct a standard basis of an ideal of the polynomial ring R [ X ] = R [ x 1 , . . . , x k ] over commutative Artinian chain ring R , which generalises a Gröbner base of a polynomial ideal over fields. We adopt the notion of the leading term of a polynomial suggested by D. A. Mikhailov and A. A. Nechaev, but using the simplification schemes introduced by V. N. Latyshev. We prove that any canonical generating system constructed by D. A. Mikhailov and A. A. Nechaev is a standard basis of the special form. We give an algorithm (based on the notion of S -polynomial) which constructs standard bases and canonical generating systems of an ideal. We define minimal and reduced standard bases and give their characterisations. We prove that a Gröbner base χ of a polynomial ideal over the field = R / rad( R ) can be lifted to a standard basis of the same cardinality over R with respect to the natural epimorphism ν : R [ X ] → [ X ] if and only if there is an ideal I R [ X ] such that I is a free R -module and Ī = (χ).

ABSTRACT Let be an integral domain and let be a prime ideal of such that every ideal not in is -invertible. We will call such an integral domain a pinched-Krull domain at . In this paper we give some characterizations of a pinched-Krull domain which are analogs of a Krull domain. We also show that is a pinched-Krull domain at if and only if is a pinched-Krull domain at ; and that is a Krull domain if and only if is a pinched-Krull domain at . Let be a nonzero maximal ideal of an integral domain , the natural ring epimorphism, a proper subring of , and . It is proved that if and are Krull domains, then is a pinched-Krull domain at . As an application we give non-Mori domains whose prime -ideals are of finite type. At the end, we study a pinched-Krull domain at a divided prime ideal.