On the tolerance lattice of tolerance factors

Grygiel, Joanna; Radelecki, Sándor

doi:10.1007/s10474-013-0340-x

Cited by 4 publications

(5 citation statements)

References 11 publications

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“…However, even in the case of lattices, many properties typical for quotient structures are not, in general, valid for factor structures. Some of them, like the homomorphism theorem and the second isomorphism theorem for lattice congruences, can be imitated in the set of all tolerances partially ordered by a particular restriction of a regular (i.e., inclusion) order of tolerances (see [20]). The resulted poset is not always a lattice, but it can be converted into a specific commutative join-directoid.…”

Section: Covering Systemsmentioning

confidence: 99%

Many Faces of Lattice Tolerances

Grygiel

2019

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Section: Covering Systemsmentioning

confidence: 99%

Many Faces of Lattice Tolerances

Grygiel

2019

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“…Let us note that results on tolerances on algebras can be found in the monograph [1], see also [2,3] and [8]. It is worth noticing that a similar attempt for defining congruences on posets was introduced and treated in [4].…”

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“…We are going to prove an analogous result also for posets. The poset (Con P, ⊆) is depicted in Figure 6: J. Grygiel and S. Radeleczki ( [8]) showed that a partial order relation ≤ on the set Tol L of tolerances on a lattice L can be introduced in such a way that for S, T ∈ Tol L with S ≤ T a tolerance T /S on the quotient lattice L/S can be defined such that the Isomorphism Theorem for tolerances (L/S)/(T /S) ∼ = L/T holds. For posets, we proceed as follows.…”

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confidence: 99%

Tolerances on posets

Chajda¹,

Länger²

2023

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The concept of a tolerance relation, shortly called tolerance, was studied on various algebras since the seventies of the twentieth century by B. Zelinka and the first author (see e.g. [6] and the monograph [1] and the references therein). Since tolerances need not be transitive, their blocks may overlap and hence in general the set of all blocks of a tolerance cannot be converted into a quotient algebra in the same way as in the case of congruences. However, G. Czédli ([7]) showed that lattices can be factorized by means of tolerances in a natural way, and J. Grygiel and S. Radeleczki ([8]) proved some variant of an Isomorphism Theorem for tolerances on lattices. The aim of the present paper is to extend the concept of a tolerance on a lattice to posets in such a way that results similar to those obtained for tolerances on lattices can be derived.

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“…Let us note that results on tolerances on algebras can be found in the monograph [1], see also [2], [3] and [8]. It is worth noticing that a similar attempt for defining congruences on posets was introduced and treated in [4].…”

mentioning

confidence: 99%

“…J. Grygiel and S. Radelecki ( [8]) showed that a partial order relation ≤ on the set Tol L of tolerances on a lattice L can be introduced in such a way that for S, T ∈ Tol L with S ≤ T a tolerance T /S on the quotient lattice L/S can be defined such that the Isomorphism Theorem for tolerances (L/S)/(T /S) ∼ = L/T holds. For posets, we proceed as follows.…”

mentioning

confidence: 99%

Tolerances on posets

Chajda¹,

Länger²

2021

Preprint

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The concept of a tolerance relation, shortly called tolerance, was studied on various algebras since the seventieth of the twentieth century by B. Zelinka and the first author (see e.g. [6] and the monograph [1] and the references therein). Since tolerances need not be transitive, their blocks may overlap and hence in general the set of all blocks of a tolerance cannot be converted into a quotient algebra in the same way as in the case of congruences. However, G. Czédli ([6]) showed that lattices can be factorized by means of tolerances in a natural way, and J. Grygiel and S. Radelecki ([8]) proved some variant of an Isomorphism Theorem for tolerances on lattices. The aim of the present paper is to extend the concept of a tolerance on a lattice to posets in such a way that results similar to those obtained for tolerances on lattices can be derived.

show abstract

On the tolerance lattice of tolerance factors

Cited by 4 publications

References 11 publications

Many Faces of Lattice Tolerances

Many Faces of Lattice Tolerances

Tolerances on posets

Tolerances on posets

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