On the arithmetic of arithmetical congruence monoids

M. Banister; J. Chaika; S. T. Chapman; W. Meyerson

Displaying similar documents to “On the arithmetic of arithmetical congruence monoids”

Unified-like product of monoids and its regularity property

Esra Kırmızı Çetinalp (2024)

Czechoslovak Mathematical Journal

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We first define a new monoid construction (called unified-like product $O ⋄_{Ω} J$ ) under a unified product $O ⋈ J$ and the Schützenberger product $O ⋄ J$ . We investigate whether this algebraic construction defined with operations of the unified and Schützenberger product specifies a monoid or not. Then, we obtain a presentation of this new product for any two monoids. Finally, we define the necessary and sufficient conditions for $O ⋄_{Ω} J$ to be regular.

The minimal closed monoids for the Galois connection $End$ - $Con$

Danica Jakubíková-Studenovská, Reinhard Pöschel, Sándor Radelecki (2024)

Mathematica Bohemica

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The minimal nontrivial endomorphism monoids $M = End Con (A, F)$ of congruence lattices of algebras $(A, F)$ defined on a finite set $A$ are described. They correspond (via the Galois connection $End$ - $Con$ ) to the maximal nontrivial congruence lattices $Con (A, F)$ investigated and characterized by the authors in previous papers. Analogous results are provided for endomorphism monoids of quasiorder lattices $Quord (A, F)$ .

The existence of solutions of the generalized pseudoprime congruence $a^{f (n)} \equiv b^{f (n)} (m o d n)$

Wayne L. McDaniel (1990)

Colloquium Mathematicae

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On the construction and the realization of wild monoids

Pavel Růžička (2018)

Archivum Mathematicum

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We develop elementary methods of computing the monoid $𝒱 (R)$ for a directly-finite regular ring $R$ . We construct a class of directly finite non-cancellative refinement monoids and realize them by regular algebras over an arbitrary field.

On a linear homogeneous congruence

A. Schinzel, M. Zakarczemny (2006)

Colloquium Mathematicae

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The number of solutions of the congruence $a ₁ x ₁ + \dots + a_{k} x_{k} \equiv 0 (m o d n)$ in the box $0 \leq x_{i} \leq b_{i}$ is estimated from below in the best possible way, provided for all i,j either $(a_{i}, n) | (a_{j}, n)$ or $(a_{j}, n) | (a_{i}, n)$ or $n | [a_{i}, a_{j}]$ .

Congruence preserving operations on the ring $ℤ_{p^{3}}$

Cyril Gavala, Miroslav Ploščica, Ivana Varga (2023)

Mathematica Bohemica

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We investigate the interval $I (p^{3})$ in the lattice of clones on the ring $ℤ_{p^{3}}$ between the clone of polynomial operations and the clone of congruence preserving operations. All clones in this interval are known and described by means of generators. In this paper, we characterize each of these clones by the property of preserving a small set of relations. These relations turn out to be in a close connection to commutators.

Congruences for Wolstenholme primes

Romeo Meštrović (2015)

Czechoslovak Mathematical Journal

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A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence $(\binom{2 p - 1}{p - 1}) \equiv 1 (mod p^{4})$ . For such a prime $p$ , we establish an expression for $(\binom{2 p - 1}{p - 1}) (mod p^{8})$ given in terms of the sums $R_{i} : = \sum_{k = 1}^{p - 1} 1 / k^{i}$ ( $i = 1, 2, 3, 4, 5, 6)$ . Further, the expression in this congruence is reduced in terms of the sums $R_{i}$ ( $i = 1, 3, 4, 5$ ). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $(\binom{2 p - 1}{p - 1}) \equiv 1 - 2 p \sum_{k = 1}^{p - 1} \frac{1}{k} - 2 p^{2} \sum_{k = 1}^{p - 1} \frac{1}{k^{2}} (mod p^{7}) .$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying...

The set of minimal distances in Krull monoids

Alfred Geroldinger, Qinghai Zhong (2016)

Acta Arithmetica

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Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say $a = u_{1} \cdot . . . \cdot u_{k}$ . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds...

Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu, Ranjan Bera, Balasubramanian Sury (2024)

Czechoslovak Mathematical Journal

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We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences $\sum_{i = 1}^{k} a_{i} x_{i} \equiv b (mod n)$ . In particular, we obtain explicit expressions for the number of solutions, where $x_{i}$ ’s are squares modulo $n$ . In addition, we obtain expressions for the number of solutions with order restrictions $x_{1} \geq \dots \geq x_{k}$ or with strict order restrictions $x_{1} > \dots > x_{k}$ in some special cases. In these results, the expressions for the number of solutions involve...

On the lattice of congruences on inverse semirings

Anwesha Bhuniya, Anjan Kumar Bhuniya (2008)

Discussiones Mathematicae - General Algebra and Applications

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Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences $ρ_{m i n}, ρ_{m a x}, ρ^{m i n}$ and $ρ^{m a x}$ on S and showed that $ρ θ = [ρ_{m i n}, ρ_{m a x}]$ and $ρ κ = [ρ^{m i n}, ρ^{m a x}]$ . Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if $ρ_{m a x}$ is a distributive lattice congruence and $ρ^{m a x}$ is a skew-ring congruence on S. If η (σ) is the...

On the quartic character of quadratic units

Zhi-Hong Sun (2013)

Acta Arithmetica

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Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, $d = 2^{r} d ₀$ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine ${(b + \sqrt (b ² + 4^{α}) / 2)}^{(p - 1) / 4)} (m o d p)$ for p = x²+(b²+4α)y² (b,x,y ∈ ℤ, 2∤b), and ${(2 a + \sqrt 4 a ² + 1)}^{(p - 1) / 4} (m o d p)$ for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for $U_{(p - 1) / 4} (m o d p)$ and the criterion for $p | U_{(p - 1) / 8}$ (if p ≡ 1 (mod 8)), where Uₙ is the Lucas sequence given by U₀ = 0, U₁ = 1 and...

Principal congruence link complements

Mark D. Baker, Alan W. Reid (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper we study principal congruence link complements in $S^{3}$ . It is known that there are only finitely many such link complements, and we make a start on enumerating them using a combination of theoretical methods and computer calculations with MAGMA.