A property of meets in slim semimodular lattices and its application to retracts

Czédli, Gábor

doi:10.1007/s44146-022-00040-z

Cited by 2 publications

(1 citation statement)

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“…the monograph [1]. Important are in particular tolerances on lattices since G. Czédli showed in [4] that every lattice L = (L, ∨, ∧) can be factorized in a natural way by any tolerance T on L, i.e. the set L/T of all blocks of T forms a lattice again.…”

Section: Introductionmentioning

confidence: 99%

Residuation in finite posets

Chajda

Länger

2021

Mathematica Slovaca

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When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left) adjointness with conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.

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Section: Introductionmentioning

confidence: 99%