On the lattice of congruences on inverse semirings

Anwesha Bhuniya; Anjan Kumar Bhuniya

Displaying similar documents to “On the lattice of congruences on inverse semirings”

Congruences for certain families of Apéry-like sequences

Zhi-Hong Sun (2022)

Czechoslovak Mathematical Journal

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We systematically investigate the expressions and congruences for both a one-parameter family ${G_{n} (x)}$ as well as a two-parameter family ${G_{n} (r, m)}$ of sequences.

New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

Ernest X. W. Xia (2015)

Colloquium Mathematicae

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Let $\bar{p p} (n)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\bar{p p} (3 n + 2) \equiv 0 (m o d 3)$ . They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for $\bar{p p} (n)$ . Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for $\bar{p p} (n)$ . Furthermore, they also constructed infinite families of congruences for $\bar{p p} (n)$ modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several...

Diamond identities for relative congruences

Gábor Czédli (1995)

Archivum Mathematicum

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For a class $K$ of structures and $A \in K$ let ${C o n}^{*} (A)$ resp. ${C o n}^{K} (A)$ denote the lattices of $*$ -congruences resp. $K$ -congruences of $A$ , cf. Weaver [25]. Let ${C o n}^{*} (K) : = I {{C o n}^{*} (A) : A \in K}$ where $I$ is the operator of forming isomorphic copies, and ${C o n}^{r} (K) : = I {{C o n}^{K} (A) : A \in K}$ . For an ordered algebra $A$ the lattice of order congruences of $A$ is denoted by ${C o n}^{<} (A)$ , and let ${C o n}^{<} (K) : = I {{C o n}^{<} (A) : A \in K}$ if $K$ is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by $Q^{s}$ and $P$ , respectively. Let $λ$ be a lattice identity and let $Σ$ be a set of lattice identities....

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.

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...

Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Zhi-Hong Sun (2011)

Acta Arithmetica

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On Alternatives of Polynomial Congruences

Mariusz Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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What should be assumed about the integral polynomials $f ₁ (x), . . ., f_{k} (x)$ in order that the solvability of the congruence $f ₁ (x) f ₂ (x) \dots f_{k} (x) \equiv 0 (m o d p)$ for sufficiently large primes p implies the solvability of the equation $f ₁ (x) f ₂ (x) \dots f_{k} (x) = 0$ in integers x? We provide some explicit characterizations for the cases when $f_{j} (x)$ are binomials or have cyclic splitting fields.

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 $n | [a_{i}, a_{j}]$ .

Congruences and homomorphisms on $Ω$ -algebras

Elijah Eghosa Edeghagba, Branimir Šešelja, Andreja Tepavčević (2017)

Kybernetika

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The topic of the paper are $Ω$ -algebras, where $Ω$ is a complete lattice. In this research we deal with congruences and homomorphisms. An $Ω$ -algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an $Ω$ -valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce $Ω$ -valued congruences, corresponding quotient $Ω$ -algebras and $Ω$ -homomorphisms and we investigate connections among these notions....

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Ernest X. W. Xia (2016)

Colloquium Mathematicae

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Let $p ̅_{o} (n)$ denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function $p ̅_{o} (n)$ have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for $p ̅_{o} (n)$ modulo 3. For example, we prove that for n, α ≥ 0, $p ̅_{o} (4^{α} (24 n + 17)) \equiv p ̅_{o} (4^{α} (24 n + 23)) \equiv 0 (m o d 3)$ .

On congruence permutable $G$ -sets

Attila Nagy (2020)

Commentationes Mathematicae Universitatis Carolinae

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An algebraic structure is said to be congruence permutable if its arbitrary congruences $α$ and $β$ satisfy the equation $α \circ β = β \circ α$ , where $\circ$ denotes the usual composition of binary relations. To an arbitrary $G$ -set $X$ satisfying $G \cap X = \emptyset$ , we assign a semigroup $(G, X, 0)$ on the base set $G \cup X \cup {0}$ containing a zero element $0 \notin G \cup X$ , and examine the connection between the congruence permutability of the $G$ -set $X$ and the semigroup $(G, X, 0)$ .