scholarly journals Residuated operators in complemented posets

2018 ◽  
Vol 11 (06) ◽  
pp. 1850097 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Unary Operation ◽  
Orthomodular Poset ◽  
Orthomodular Lattices ◽  
Orthomodular Posets ◽  
Residuated Poset ◽  
Pseudocomplemented Poset ◽  
Boolean Poset

Using the operators of taking upper and lower cones in a poset with a unary operation, we define operators [Formula: see text] and [Formula: see text] in the sense of multiplication and residuation, respectively, and we show that by using these operators, a general modification of residuation can be introduced. A relatively pseudocomplemented poset can be considered as a prototype of such an operator residuated poset. As main results, we prove that every Boolean poset as well as every pseudo-orthomodular poset can be organized into a (left) operator residuated structure. Some results on pseudo-orthomodular posets are presented which show the analogy to orthomodular lattices and orthomodular posets.

How to introduce the connective implication in orthomodular posets

2020 ◽  
pp. 2150066
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Orthomodular Poset ◽  
Natural Question ◽  
Partial Operation ◽  
Orthomodular Posets ◽  
Residuated Poset ◽  
One To One ◽  
Additional Assumptions

Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be introduced in such a way that it is related with conjunction, i.e. with the partial operation meet, by means of some kind of adjointness. We present here such an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptions, such an unsharp residuated poset can be converted into an orthomodular poset and that this assignment is nearly one-to-one.

Sublattices and Δ-blocks of orthomodular posets

2020 ◽  
Vol 30 (7) ◽  
pp. 1401-1423
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Hilbert Space ◽  
Binary Relation ◽  
Complete Lattice ◽  
Quantum Systems ◽  
Orthomodular Poset ◽  
Orthomodular Lattices ◽  
Binary Operator ◽  
Orthomodular Posets ◽  
Finite Set

Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$-blocks with respect to Boolean subalgebras and distributive subposets they include.

scholarly journals A partial Baer *-semigroup of relations

1975 ◽  
Vol 19 (2) ◽  
pp. 160-172
Author(s):  
R. H. Schelp
Keyword(s):  
Partially Ordered Sets ◽  
Boolean Lattice ◽  
Orthomodular Poset ◽  
Ordered Sets ◽  
Orthomodular Lattices ◽  
Partially Ordered ◽  
Orthomodular Posets ◽  
The Family ◽  
Boolean Lattices ◽  
Atomic Lattices

It is shown in Gudder and Schelp (1970) that partial Baer *-semigroups coordinatize orthomodular partially ordered sets (orthomodular posets). This means for P an orthomodular poset there exists a partial Baer *-semigroup whose closed projections are order isomorphic to P preserving ortho-complementation. This coordinatization theorem generalizes Foulis (1960) in which orthomodular lattices are coordinatized by Baer *-semigroups. In particular Foulis (unpublished) shows that any complete atomic Boolean lattice is coordinatized by a Bear *-semigroup of relations. Since Greechie (1968), (1971) shows that a whole class of orthomodular posets can be formed by “pasting” together Boolean lattices, it is natural to consider the following problem. Let y be a family of Baer *-semigroups of relations which coordinatize the family B of complete atomic lattices. Is it possible to construct a partial *-semigroup of relations R which contains each member of Y such that when P is an orthomodular poset obtained by a “Greechie pasting” of members of 38 then 91 coordinatizes R This question is considered in the sequel and answered affirmatively for a certain subclass of “Greechie pasted” orthomodular posets. In addition the construction of 8)t nicely fulfills another objective in that it provides us with “nontrivial” coordinate partial Baer *-semigroups for a whole family of well known orthomodular posets. This is particularly significant since the only other known coordinate partial Baer *-semigroups, for those posets in this family which are not lattices, are the “minimal” ones given in Gudderand and Schelp (1970).

scholarly journals The logic of orthomodular posets of finite height

2020 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Positive Answer ◽  
Complete List ◽  
Orthomodular Poset ◽  
Central Question ◽  
Implication Operator ◽  
Algebraizable Logic ◽  
Binary Operator ◽  
Orthomodular Posets ◽  
Finite Height

Abstract Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. A central question is how to introduce implication in such a logic. We give a positive answer whenever the orthomodular poset in question is of finite height. The crucial advantage of our solution is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and that its implication operator is everywhere defined. We present here a complete list of axioms for implication orthomodular posets. This enables us to derive an axiomatization in Gentzen style for the algebraizable logic of orthomodular posets of finite height.

Weakly orthomodular and dually weakly orthomodular posets

2018 ◽  
Vol 11 (02) ◽  
pp. 1850093 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Orthomodular Poset ◽  
Algebraic Semantic ◽  
Orthomodular Posets ◽  
Congruence Properties

Orthomodular posets form an algebraic semantic for the logic of quantum mechanics. We show several methods how to construct orthomodular posets via a representation within the powerset of a given set. Further, we generalize this concept to the concept of weakly orthomodular and dually weakly orthomodular posets where the complementation need not be antitone or an involution. We show several interesting examples of such posets and prove which intervals of these posets are weakly orthomodular or dually weakly orthomodular again. To every (dually) weakly orthomodular poset can be assigned an algebra with total operations, a so-called (dually) weakly orthomodular [Formula: see text]-lattice. We study properties of these [Formula: see text]-lattices and show that the variety of these [Formula: see text]-lattices has nice congruence properties.

scholarly journals Left residuated lattices induced by lattices with a unary operation

2019 ◽  
Vol 24 (2) ◽  
pp. 723-729
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Unary Operation ◽  
Residuated Lattices ◽  
Boolean Algebras ◽  
Natural Question ◽  
Similar Technique ◽  
Orthomodular Lattices

Abstract In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

An axiomatisation of quantum logic

1973 ◽  
Vol 38 (3) ◽  
pp. 389-392 ◽  
Author(s):  
Ian D. Clark
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Binary Operation ◽  
Ordered Set ◽  
Unary Operation ◽  
Axiom System ◽  
Partial Ordering ◽  
Orthomodular Poset ◽  
Implication Algebra ◽  
Image Position

The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset.In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa.Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following:(1) the greatest lower bound of a and Na exists and is 0,(2) a ≤ b implies Nb ≤ Na,(3) NNa = a.Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfyingwhere a ≤ b means a ∧ b = a, as usual.Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following:An orthomodular lattice gives a Finch implication algebra by defining → byA Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ byThe orthocomplementation is unchanged in both cases.

The zero divisor graphs of Boolean posets

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Vinayak Joshi ◽  
Anagha Khiste
Keyword(s):  
Zero Divisor ◽  
Pseudocomplemented Poset ◽  
Boolean Poset

AbstractIn this paper, it is proved that if B is a Boolean poset and S is a bounded pseudocomplemented poset such that S\Z(S) = {1}, then Γ(B) ≌ Γ(S) if and only if B ≌ S. Further, we characterize the graphs which can be realized as zero divisor graphs of Boolean posets.

scholarly journals A note on constructible lattices

1973 ◽  
Vol 15 (3) ◽  
pp. 296-297 ◽  
Author(s):  
D. H. Adams
Keyword(s):  
Orthomodular Lattices ◽  
Orthomodular Posets ◽  
Boolean Lattices ◽  
Subdirect Products

It is our purpose to show that the constructible orthomodular lattices defined by Janowitz in [2], are embeddable into Boolean lattices. In fact they are subdirect products of Boolean lattices, where the subdirect products are taken in the class of orthomodular posets. We shall make these notions precise. Other concepts, such as disjoint sum and constructible lattice, are defined in [1] and [2].

Roughness in Orthomodular Lattices

2012 ◽  
Vol 2 (1) ◽  
pp. 9
Author(s):  
E. K. R Nagarajan ◽  
D. Umadevi
Keyword(s):  
Orthomodular Lattices
Sign in / Sign up

Export Citation Format

Share Document