scholarly journals Finite pseudocomplemented lattices and ‘permutoedre’

1993 ◽  
Vol 111 (1-3) ◽  
pp. 105-112 ◽  
Author(s):  
C. Chameni Nembua ◽  
B. Monjardet
Keyword(s):  
Pseudocomplemented Lattices

scholarly journals Filters and congruences in sectionally pseudocomplemented lattices and posets

2021 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Algebraic Semantics ◽  
Basic Properties ◽  
Algebraic Constructions ◽  
Pseudocomplemented Lattices ◽  
Intuitionistic Logics

AbstractTogether with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that—similar to relatively pseudocomplemented lattices—these structures can serve as an algebraic semantics of certain intuitionistic logics. The aim of the present paper is to define congruences and filters in these structures, derive mutual relationships between them and describe basic properties of congruences in strongly sectionally pseudocomplemented posets. For the description of filters in both sectionally pseudocomplemented lattices and posets, we use the tools introduced by A. Ursini, i.e., ideal terms and the closedness with respect to them. It seems to be of some interest that a similar machinery can be applied also for strongly sectionally pseudocomplemented posets in spite of the fact that the corresponding ideal terms are not everywhere defined.

scholarly journals Bounded endomorphisms of free P-algebras

1992 ◽  
Vol 34 (2) ◽  
pp. 209-214
Author(s):  
Daniel Ševčovič
Keyword(s):  
Free Algebra ◽  
Present Note ◽  
Finitely Generated ◽  
Efficient Tool ◽  
Pseudocomplemented Lattices

The present note deals with bounded endomorphisms of free p-algebras (pseudocomplemented lattices). The idea of bounded homomorphisms was introduced by R. McKenzie in [8]. T. Katriňák [5] subsequently studied the properties of bounded homomorphisms for the varieties of p-algebras. This concept is also an efficient tool for the characterization of, so-called, splitting as well as projective algebras in the varieties of all lattices or p-algebras. For details the reader is referred to [2], [5], [6], [7] and other references therein. Let us emphasize that the main results that are contained in the above mentioned references strongly depend on the boundedness of each endomorphism of any finitely generated free algebra in a given variety.

π-Complemented Algebras Through Pseudocomplemented Lattices

Order ◽  
2011 ◽  
Vol 29 (3) ◽  
pp. 463-479 ◽  
Author(s):  
Juan Carlos Cabello ◽  
Miguel Cabrera ◽  
Antonio Fernández López
Keyword(s):  
Pseudocomplemented Lattices

scholarly journals Stone Lattices

2015 ◽  
Vol 23 (4) ◽  
pp. 387-396 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Lattice Theory ◽  
Cartesian Product ◽  
Unary Operation ◽  
Binary Operations ◽  
Pseudocomplemented Lattice ◽  
Stone Lattice ◽  
Current Article ◽  
Boolean Lattices ◽  
Pseudocomplemented Lattices ◽  
Natural Way

Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].

scholarly journals Sectionally Pseudocomplemented Residual Lattice

1970 ◽  
Vol 6 (2) ◽  
pp. 53-54
Author(s):  
MD Zaidur Rahman ◽  
Md Abdul Kalam Azad ◽  
Md Nazmul Hasan
Keyword(s):  
Key Words ◽  
Basic Concept ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Commutative Monoid ◽  
Bounded Lattice ◽  
Complemented Lattices ◽  
Pseudocomplemented Lattices

At first, we recall the basic concept, By a residual lattice is meant an algebra L = (L,∨,∧,∗,o,0,1) such that (i) L = (L,∨,∧,0,1) is a bounded lattice, (ii) L = (L,∗,1) is a commutative monoid, (iii) it satisfies the so-called adjoin ness property: (x ∨ y) ∗ z = y if and only if y ≤ z ≤ x o y Let us note [7] that x ∨ y is the greatest element of the set (x ∨ y) ∗ z = y Moreover, if we consider x ∗ y = x ∧ y , then x o y is the relative pseudo-complement of x with respect to y, i. e., for ∗ = ∧ residuated lattices are just relatively pseudo-complemented lattices. The identities characterizing sectionally pseudocomplemented lattices are presented in [3] i.e. the class of these lattices is a variety in the signature {∨,∧,o,1}. We are going to apply a similar approach for the adjointness property: Key words: Residuated lattice; non Distributive; Residuated Abeliean; commutative monoid: DOI: http://dx.doi.org/10.3329/diujst.v6i2.9345 DIUJST 2011; 6(2): 53-54

Finite pseudocomplemented lattices: The spectra and the Glivenko congruence

2011 ◽  
Vol 66 (1-2) ◽  
pp. 151-161 ◽  
Author(s):  
T. Katriňák ◽  
J. Guričan
Keyword(s):  
Pseudocomplemented Lattices

scholarly journals Relatively residuated lattices and posets

2020 ◽  
Vol 70 (2) ◽  
pp. 239-250
Author(s):  
Ivan Chajda ◽  
Jan Kühr ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Unary Operation ◽  
General Concept ◽  
Residuated Lattices ◽  
Distributive Lattices ◽  
Pseudocomplemented Lattice ◽  
Pseudocomplemented Lattices

Abstract It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relatively residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relatively residuated lattices which are similar to those known for residuated ones and extend our results to posets.

scholarly journals Sectionally Pseudocomplemented Posets

Order ◽  
2021 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger ◽  
Jan Paseka
Keyword(s):  
Modular Lattice ◽  
Distributive Lattices ◽  
Macneille Completion ◽  
Ordinal Sum ◽  
Pseudocomplemented Lattice ◽  
Pseudocomplemented Poset ◽  
Pseudocomplemented Lattices ◽  
Congruence Properties

AbstractThe 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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