Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Chajda, Ivan; Länger, Helmut

doi:10.3390/sym13050753

Cited by 5 publications

(3 citation statements)

References 13 publications

(21 reference statements)

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“…The authors already published papers treating this topic, see e.g. [4], [7] and [8]. A characterization of pseudocomplemented posets by triples similar to that for lattices mentioned above was settled for particular cases by the first author and R. Halaš in [3].…”

Section: Introductionmentioning

confidence: 99%

Sectionally Pseudocomplemented Posets

Chajda

Länger

Paseka

2021

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The concept of a sectionally pseudocomplemented lattice was introduced in Birkhoff (1979) as an extension of relative pseudocomplementation for not necessarily distributive lattices. The typical example of such a lattice is the non-modular lattice N5. The aim of this paper is to extend the concept of sectional pseudocomplementation from lattices to posets. At first we show that the class of sectionally pseudocomplemented lattices forms a variety of lattices which can be described by two simple identities. This variety has nice congruence properties. We summarize properties of sectionally pseudocomplemented posets and show differences to relative pseudocomplementation. We prove that every sectionally pseudocomplemented poset is completely L-semidistributive. We introduce the concept of congruence on these posets and show when the quotient structure becomes a poset again. Finally, we study the Dedekind-MacNeille completion of sectionally pseudocomplemented posets. We show that contrary to the case of relatively pseudocomplemented posets, this completion need not be sectionally pseudocomplemented but we present the construction of a so-called generalized ordinal sum which enables us to construct the Dedekind-MacNeille completion provided the completions of the summands are known.

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Section: Introductionmentioning

confidence: 99%

Sectionally Pseudocomplemented Posets

Chajda

Länger

Paseka

2021

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“…If (S, ∧, * , 0) is even a lattice then it is called a Heyting algebra, see [14] and [17]. For posets the concept of pseudocomplementation was extended and studied by the authors in [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

The logic with unsharp implication and negation

Chajda¹,

Länger²

2023

Preprint

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It is well-known that intuitionistic logics can be formalized by means of Brouwerian semilattices, i.e. relatively pseudocomplemented semilattices. Then the logical connective implication is considered to be the relative pseudocomplement and conjunction is the semilattice operation meet. If the Brouwerian semilattice has a bottom element 0 then the relative pseudocomplement with respect to 0 is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the Ascending Chain Condition, which is trivially satisfied in finite semilattices, and introduce the connective negation x 0 as the set of all maximal elements z satisfying x ∧ z = 0 and the connective implication x → y as the set of all maximal elements z satisfying x ∧ z ≤ y. Such a negation and implication is "unsharp" since it assigns to one entry x or to two entries x and y belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication, respectively, still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. Several examples are presented. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.

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“…(A variant of this meaning of the term should be mentioned: in some papers, e.g., [7,9], a lattice is said to be sectionally pseudocomplemented if the pseudocomplement of a ∨ b in the principal filter [a) exists for all a, b. This view on sectional complementedness has been quite recently generalized in [5,7] for posets. )…”

Section: Introductionmentioning

confidence: 99%

Extension of sectional pseudocomplementation in posets

Cı̄rulis

2022

Preprint

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Sectional pseudocomplementation (sp-complementation) on a poset is a partial operation * which associates with every pair (x,y) of elements, where x ≥ y, the pseudocomplement x*y of x in the upper section [y). Any total extension → of * is said to be an extended sp-complementation and is considered as an implication-like operation. Extended sp-complementations have already be studied on semilattices and lattices. We describe several naturally arising classes of general posets with extended sp-complementation, present respective elementary properties of this operation, demonstrate that two other known attempts to isolate particular such classes are in fact not quite correct, and suggest suitable improvements.MSC Classification: 03G25 , 06A99 , 08A99

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Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Cited by 5 publications

References 13 publications

Sectionally Pseudocomplemented Posets

Sectionally Pseudocomplemented Posets

The logic with unsharp implication and negation

Extension of sectional pseudocomplementation in posets

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