Diamond identities for relative congruences

Gábor Czédli

Displaying similar documents to “Diamond identities for relative congruences”

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...

Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

Gábor Czédli (2024)

Mathematica Bohemica

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Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_{3}$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset $P$ is said to be JConSPS-representable if there is an SPS lattice $L$ such that $P$ is isomorphic to the poset $J (Con L)$ of join-irreducible congruences of $L$ . We prove that...

2-normalization of lattices

Ivan Chajda, W. Cheng, S. L. Wismath (2008)

Czechoslovak Mathematical Journal

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Let $τ$ be a type of algebras. A valuation of terms of type $τ$ is a function $v$ assigning to each term $t$ of type $τ$ a value $v (t) \geq 0$ . For $k \geq 1$ , an identity $s \approx t$ of type $τ$ is said to be $k$ -normal (with respect to valuation $v$ ) if either $s = t$ or both $s$ and $t$ have value $\geq k$ . Taking $k = 1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$ -normal (with respect to the valuation $v$ ) if all its identities are $k$ -normal. For any variety $V$ , there...

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....

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...

A class of multiplicative lattices

Tiberiu Dumitrescu, Mihai Epure (2021)

Czechoslovak Mathematical Journal

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We study the multiplicative lattices $L$ which satisfy the condition $a = (a : (a : b)) (a : b)$ for all $a, b \in L$ . Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group $ℤ$ or $ℝ$ . A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.

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...

Hyperreflexivity of bilattices

Kamila Kliś-Garlicka (2016)

Czechoslovak Mathematical Journal

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The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice $ℒ$ we can construct the bilattice $Σ_{ℒ}$ . Similarly, having a bilattice $Σ$ we may consider the lattice $ℒ_{Σ}$ . In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples...