Minimal bounded lattices with an antitone involution the complemented elements of which do not form a sublattice

2008 ◽  
Vol 28 (2) ◽  
pp. 251
Author(s):  
Ivan Chajda ◽  
Helmut Langer
Keyword(s):  
Antitone Involution

scholarly journals QUANTIFIERS ON LATTICES WITH AN ANTITONE INVOLUTION

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Antitone Involution

scholarly journals Quantifiers on lattices with an antitone involution

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Antitone Involution

AbstractQuantifiers on lattices with an antitone involution are considered and it is proved that the poset of existential quantifiers is antiisomorphic to the poset of relatively complete sublattices.

Symmetric Differences on Posets with an Antitone Involution

Order ◽  
2011 ◽  
Vol 29 (1) ◽  
pp. 215-225
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Antitone Involution

scholarly journals Adjoint Operations in Twist-Products of Lattices

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 253
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Sufficient Conditions ◽  
Residuated Lattices ◽  
Double Negation ◽  
Antitone Involution ◽  
Binary Operations ◽  
Double Negation Law ◽  
Commutative Residuated Lattice

Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.

An approach to orthomodular lattices via lattices with an antitone involution

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Ivan Chajda ◽  
Sándor Radeleczki
Keyword(s):  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Bounded Lattice ◽  
Antitone Involution ◽  
Weakly Orthomodular Lattice

AbstractSeveral characterizations of orthomodular lattices based on properties of an antitone involution or on sectional antitone involutions are given. Another approach is based on properties of the implication operation which can be derived in every orthomodular lattice but can be also added to basic operations of a bounded lattice. Finally, we introduce the so-called weakly orthomodular lattice as a generalization of orthomodular lattices.

Polynomial permutations on bounded commutative directoids with an antitone involution

2010 ◽  
Vol 15 (1) ◽  
pp. 183-186
Author(s):  
Ivan Chajda ◽  
Miroslav Kolařík ◽  
Helmut Länger
Keyword(s):  
Antitone Involution

A triple representation of lattice effect algebras

2016 ◽  
Vol 66 (6) ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Mutual Relationship ◽  
Lattice Effect Algebra ◽  
Antitone Involution ◽  
Lattice Effect ◽  
One To One ◽  
Mv Algebras

AbstractA mutual relationship between MV-algebras and coupled semirings as established by L. P. Belluce, A. Di Nola, A. R. Ferraioli and B. Gerla is extended to lattice effect algebras and so-called characterizing triples. We show that this correspondence is in fact one-to-one and hence every lattice effect algebra can be considered as an ordered triple consisting of two semiring-like structures and an antitone involution which is an isomorphism between these structures.

scholarly journals Consistent posets

2021 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Classical Logic ◽  
Macneille Completion ◽  
Algebraic Semantics ◽  
Bounded Lattice ◽  
Antitone Involution ◽  
Bounded Posets

AbstractWe introduce so-called consistent posets which are bounded posets with an antitone involution $$'$$ ′ where the lower cones of $$x,x'$$ x , x ′ and of $$y,y'$$ y , y ′ coincide provided that x, y are different from 0, 1 and, moreover, if x, y are different from 0, then their lower cone is different from 0, too. We show that these posets can be represented by means of commutative meet-directoids with an antitone involution satisfying certain identities and implications. In the case of a finite distributive or strongly modular consistent poset, this poset can be converted into a residuated structure and hence it can serve as an algebraic semantics of a certain non-classical logic with unsharp conjunction and implication. Finally we show that the Dedekind–MacNeille completion of a consistent poset is a consistent lattice, i.e., a bounded lattice with an antitone involution satisfying the above-mentioned properties.

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