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Note on quasivarieties generated by finite pointed abelian groups

  • Ainur Basheyeva and Svetlana Lutsak EMAIL logo
Published/Copyright: March 21, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

We prove that a finite pointed abelian group generates a finitely axiomatizable variety that has a finite quasivariety lattice. As a consequence, we obtain that a quasivariety generated by a finite pointed abelian group has a finite basis of quasi-identities. The problems arising from the results obtained are also discussed.

Keywords: variety; quasivariety; quasivariety lattice; finitely axiomatizable class; abelian group; pointed group
MSC 2010: 06B20; 08B05; 08C15; 20N99

1 Introduction

A pointed algebra is obtained from the given algebra by adding a finite set of constants (nullary operations) into the signature. It is also called a pointed enrichment of the given algebra. For example, a pointed abelian group is a pair ( G , g ) consisting of an abelian group G together with a distinguished element g of G . The equational and quasi-equational theories of an algebra and their pointed enrichments can have completely different properties. Indeed, the remarkable theorem of group theory due to Oates and Powell [1] states that every finite group has a finite basis of identities. On the other hand, Bryant [2] constructed a finite pointed group that has no finite basis of identities. For the quasi-equational theories, similar example was found among finite lattices. Namely, there is a finite lattice that has a finite basis of quasi-identities; however, there exists its pointed enrichment that has no finite basis of quasi-identities [3]. Moreover, the aforementioned lattice generates a variety with a finite lattice of quasivarieties, and at the same time, there is a pointed enrichment of it, generating a variety with an infinite lattice of quasivarieties. From the aforementioned case naturally arises the question: Which (quasi-)equational properties of finite algebras (groups, lattices) are preserved with respect to pointed enrichments? (cf. [4]). Note that the finite basis problem for identities of finite pointed groups was discussed in [2] and is still open.

Here, we will examine the aforementioned problem for finite abelian groups and their pointed enrichments. Namely, we establish that a finite pointed abelian group generates a finitely axiomatizable variety that has a finite quasivariety lattice. As a consequence, we obtain that the quasivariety generated by a finite pointed abelian group has a finite basis of quasi-identities. We would like to note that the latter fact was proved in [4], where the authors used a specific property of being a directed representable quasivariety. In our approach, we show the use of a finiteness of quasivariety lattices, which can be applied for a wider class of quasivarieties.

2 Basic definitions and preliminaries

Let us briefly recall some main definitions. For more information on the basic notions of universal algebra and group theory used throughout this article, we refer to [5–7].

A pointed group is an algebra ⟨ G ; ⋅ , − 1 , 1 , C ⟩ of the signature ⟨ ⋅ , − 1 , 1 , C ⟩ , where ⟨ ⋅ , − 1 , 1 ⟩ -reduct is a group and C is a finite set of constants. A pointed group ⟨ G ; ⋅ , − 1 , 1 , C ⟩ is also called a pointed enrichment of the group ⟨ G ; ⋅ , − 1 , 1 ⟩ and is denoted by G c . A pointed group is abelian if its group’s reduct is an abelian group.

A quasivariety is a class of similar algebras that is closed with respect to subalgebras, direct products, and ultraproducts. Equivalently, a quasivariety is the same thing as a class of similar algebras axiomatized by a set of quasi-identities. A quasi-identity means a universal Horn sentence with the non-empty positive part, which is of the form:

( ∀ x ¯ ) [ p 1 ( x ¯ ) ≈ q 1 ( x ¯ ) ∧ ⋯ ∧ p n ( x ¯ ) ≈ q n ( x ¯ ) → p ( x ¯ ) ≈ q ( x ¯ ) ] ,

where p , q , p 1 , q 1 , … , p n , q n are the terms of the signature σ . A variety is a quasivariety that is closed under homomorphisms. According to Birkhoff’s theorem, a variety is a class of similar algebras axiomatized by a set of identities, where, by an identity, we mean a sentence of the form ( ∀ x ¯ ) [ s ( x ¯ ) ≈ t ( x ¯ ) ] for some terms s ( x ¯ ) and t ( x ¯ ) .

A quasivariety ℛ ⊆ K is finitely axiomatizable relative to K if there is a finite set of quasi-identities Σ such that Mod ( Σ ) ∩ K = ℛ . In this case, Σ is a basis of quasi-identities of ℛ relative to K . An algebra has a finite basis of quasi-identities (identities) if the quasivariety (variety) generated by this algebra is finitely axiomatizable.

By Q ( K ) ( V ( K ) ), we denote the smallest quasivariety (variety) containing a class K . If K is a finite family of finite algebras, then Q ( K ) is called finitely generated. In event K = { A 1 , … , A n } , we write Q ( A 1 , … , A n ) instead of Q ( { A 1 , … , A n } ) .

For an algebra A , by Con A , we denote the congruence lattice of A . An algebra A is said to be subdirectly irreducible if the least congruence 0 A of the congruence lattice Con A is completely meet irreducible. By Birkhoff’s representation theorem [8], every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras.

For a quasivariety ℛ , by L q ( ℛ ) , we denote a quasivariety lattice of ℛ , which is the set of all subquasivarieties ordered by the set’s inclusion ⊆ . Let K be a subclass of the quasivariety ℛ and L q ( ℛ , K ) = { ℳ ∩ K ∣ ℳ ∈ L q (ℛ) } . It is easy to see that L q ( ℛ , K ) forms a complete lattice relative to the set inclusion ⊆ .

A finite algebra A is quasicritical if A is not a subdirect product of its proper subalgebras.

Lemma 1

[9, Lemma 9] Let K be the class of all quasicritical algebras of the quasivariety ℛ . Then, L q ( ℛ ) ≅ L q ( ℛ , K ) .

As a partial case of Theorem 2.13 in [10], we have

Theorem 2

[10] A finite pointed abelian group A is quasicritical if and only if a decomposition of A into a direct sum

A 1 ⊕ … ⊕ A m

possesses the following properties: A 0 , … , A m are the finite cyclic primary pointed groups of different orders p 1 n 1 , … , p m n m , and c A i = 0 A i at most for one i ≠ m ; moreover, if c A i = 0 A i , i ≤ m , then the order of the pointed group A i is the greatest among those A j for which p i = p j .

Theorem 3

(McKenzie’s theorem [11]) A locally finite congruence modular variety with a finite number of subdirectly irreducible algebras has a finite basis of identities.

3 Identities of pointed abelian groups

By the Oates-Powell theorem, a finite group has a finite basis of identities. The Braynt’s example of a pointed group that has no finite basis of identities shows that the Oates-Powell arguments generally do not work in the class of finite pointed groups. However, for pointed abelian groups, we have

Theorem 4

A finite pointed abelian group has a finite basis of identities.

Proof

Since an abelian group and any of its pointed enrichments have the same set of polynomials, by definition of congruence, every equivalence on an abelian group is a congruence if and only if it is a congruence on any (some), its pointed enrichment. It means that an abelian group and any of its pointed enrichment have the same lattices of congruences. Hence, for a finite abelian group A and its pointed enrichment A c , a congruence θ is meet irreducible in Con A if and only if θ is meet irreducible in Con A c . Since, for every meet irreducible congruence θ , the quotient group A ∕ θ is subdirectly irreducible, we obtain that a finite abelian group is subdirectly irreducible if and only if its pointed enrichment is subdirectly irreducible.

Let B c ∈ V ( A c ) and B c be subdirectly irreducible. By Birkhoff’s theorem, B ∈ HSP ( A c ) , where H ( S , P ) is an operator of taking all homomorphic images (subalgebras, direct products). Thus, there are C c ∈ SP ( A c ) and a homomorphism h : C c → B c such that ker h is meet irreducible. Hence, B is subdirectly irreducible in V ( A ) . Since a variety generated by a finite abelian group has a finite number of finite subdirectly irreducible groups, there is a finite number n > 0 such that the power of every subdirectly irreducible group in V ( A c ) is bounded by n . Hence, every subdirectly irreducible pointed group in V ( A c ) is bounded by n . Since V ( A c ) is locally finite, the number of all subdirectly irreducible pointed groups in V ( A c ) is finite. Since Con A = Con A c and a variety of abelian groups is congruence permutable, whence congruence modular, every variety of pointed abelian groups is congruence modular.

Thus, we obtain that a variety V ( A c ) is congruence modular and has a finite number of finite subdirectly irreducible pointed abelian groups. By Theorem 3, we obtain that V ( A c ) is finitely axiomatizable.□

4 Quasivariety of pointed abelian groups

By the general theory of abelian groups, a quasivariety generated by a finite abelian group is a variety, and it has a finite variety lattice that coincides with its quasivariety lattice. Also, it gives us that a finite abelian group has a finite basis of quasi-identities by the Oates-Powell theorem.

We will establish the similar results for pointed abelian groups.

First, we prove the finiteness of quasivariety lattices.

Theorem 5

A quasivariety generated by a finite set of finite pointed abelian groups has a finite quasivariety lattice.

Proof

We will prove a slightly more general result. Namely, we prove that a finitely generated variety V of pointed abelian groups has a finite quasivariety lattice. Since, for any subquasivariety ℛ of V , the quasivariety lattice L q ( ℛ ) is a sublattice of L q ( V ) , we obtain a proof of the theorem.

Let V be a variety generated by a finite set { A 1 , … , A k } of finite pointed abelian groups, i.e., V = V ( A 1 , … , A k ) . By Lemma 1, it is enough to show that V contains a finite number of quasicritical algebras.

The main theorem on the structure of finite abelian groups states that a finite abelian group is isomorphic to a direct sum of some finite primary cyclic groups [7]. By definition, a group’s reduct of a pointed abelian group is an abelian group. Hence, as the congruences of an abelian group and its pointed enrichment coincide, we obtain that a finite pointed abelian group is isomorphic to a direct sum of some finite primary pointed cyclic groups, where a primary pointed abelian group is obtained from a primary abelian group by adding a fixed finite number of constants.

Put n = ∣ A 1 ∣ × … × ∣ A k ∣ . Then, the identity ∀ x [ x n = 1 ] holds on every pointed abelian group from V . Hence, every primary pointed abelian group has at most n elements. Since the number of all primary abelian groups of power at most n is finite, we obtain that the number of all pointed abelian groups of a cardinality at most n belonging to the variety V is finite too. By Theorem 2, a finite quasicritical pointed abelian group A is a direct sum B 1 ⊕ … ⊕ B m of some finite cyclic primary pointed groups B 0 , … , B m of different orders p 1 n 1 , … , p m n m . Since the set S of all primary pointed abelian groups is finite in V ( A 1 , … , A k ) , the set of all different orders p 1 n 1 , … , p m n m has a cardinality at most 2 ∣ S ∣ , when it is finite. Thus, V contains a finite number of quasicritical pointed abelian groups.□

Furthermore, we need the following fact:

Lemma 6

Suppose that a quasivariety ℛ has a finite quasivariety lattice. Then, every subquasivariety of ℛ is finitely axiomatizable relative to ℛ .

Proof

Let K be a subquasivariety of ℛ and L q ( ℛ ) be finite. And let

S = { ℳ ∈ L q ( ℛ ) ∣ ℳ ⊈ K } .

For every ℳ ∈ S , there is a quasi-identity φ ℳ such that K ⊧ φ ℳ and ℳ ⊭ φ ℳ . Let Σ K = { φ ℳ ∣ ℳ ∈ S } . Since a quasivariety lattice of ℛ is finite, the set S is finite, whence Σ K is a finite set of quasi-identities.

We show that K = Mod ( Σ K ) ∩ ℛ , which means that K is finitely axiomatizable with respect to ℛ . Since K ⊧ Σ K and K ⊆ ℛ , K ⊆ Mod ( Σ K ) ∩ ℛ . For inverse inclusion, suppose that A ⊧ Mod ( Σ K ) ∩ ℛ , i.e., A ∈ ℛ and A ⊧ Σ K . These give us Q ( A ) ⊆ ℛ and Q ( A ) ⊧ Σ K . Assume Q ( A ) ⊈ K . It means Q ( A ) ∈ S . Hence, there is a quasi-identity φ Q ( A ) ∈ Σ K such that K ⊧ φ Q ( A ) and Q(A) ⊭ φ Q ( A ) . It contradicts to our assumption A ⊧ Σ K . Thus, Q ( A ) ⊆ K . Hence, A ∈ K , i.e., K ⊇ Mod ( Σ K ) ∩ ℛ .□

The next result is a consequence of Theorems 4 and 5 and Lemma 6.

Theorem 7

A finite pointed abelian group has a finite basis of quasi-identities.

Proof

Let A be a finite pointed abelian group. Theorems 4 and 5 provide that a variety V ( A ) is finitely axiomatizable and has a finite quasivariety lattice. By Lemma 6, a quasivariety Q ( A ) is finitely axiomatizable relative to V ( A ) . Since V ( A ) is finitely axiomatizable and Q ( A ) is finitely axiomatizable relative to V ( A ) , according to the compactness theorem, we obtain that Q ( A ) is finitely axiomatizable.□

5 Conclusion remarks

We would like to note that the results obtained about abelian groups are completely opposite to the results for finite (modular) lattices. Indeed, as we noted in Section 1, a 7-element modular lattice L and its pointed enrichment L c , which were constructed in [3], such that (a) L has a finite basis of quasi-identities but L c has no one and (b) the quasivariety lattice of V ( L ) is finite but the quasivariety lattice of V ( L c ) is infinite. These facts show that the problem “Which (quasi-)equational properties of finite algebras (groups, lattices) are preserved with respect to pointed enrichments?” has a lot of obstacles.

The following two problems come from the well-known result due to Ol’shansky [12]: a finite group has a finite basis of quasi-identities if and only if all of its nilpotent subgroups are abelian. According to this fact and Theorem 7, it is natural to ask:

Problem 1

Does there exist a finite pointed group that has no finite basis of quasi-identities and all nilpotent subgroups of its group’s reduct are abelian?

And more general:

Problem 2

Which finite pointed group (lattice) generates a finitely axiomatizable quasivariety?

As we noted in Section 1, the finite basis problem for finitely generated varieties of pointed groups is discussed in [2].

Acknowledgement

The authors thank the reviewers for useful remarks and constructive comments that made it possible to improve this manuscript.

  1. Funding information: Ainur Basheyeva is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP13268735). Svetlana Lutsak was funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP09058390).

  2. Conflict of interest: The authors state no conflicts of interest.

  3. Data availability statement: All data generated or analyzed during this study are included in this published work.

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Received: 2023-09-29
Revised: 2024-01-28
Accepted: 2024-01-31
Published Online: 2024-03-21

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0181
Keywords for this article
variety; quasivariety; quasivariety lattice; finitely axiomatizable class; abelian group; pointed group
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  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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