scholarly journals Residuation in orthomodular lattices

2017 ◽  
Vol 5 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Orthomodular Lattices ◽  
Double Negation ◽  
Converse Statement ◽  
One To One ◽  
Double Negation Law

Abstract We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.

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.

scholarly journals Residuation in non-associative MV-algebras

2018 ◽  
Vol 68 (6) ◽  
pp. 1313-1320
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Binary Operation ◽  
Residuated Lattice ◽  
Natural Question ◽  
Natural Conditions ◽  
Double Negation ◽  
Mv Algebra ◽  
Residuated Poset ◽  
Double Negation Law ◽  
Mv Algebras

Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

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.

scholarly journals When does a semiring become a residuated lattice?

2016 ◽  
Vol 09 (04) ◽  
pp. 1650088
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
The Other ◽  
Natural Question ◽  
Double Negation ◽  
Complete Distributivity ◽  
Completely Distributive ◽  
Double Negation Law ◽  
The Given ◽  
Converse Assertion

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.

Separation axioms in \(L\)-fuzzifying supra-topology

2017 ◽  
Vol 8 (1) ◽  
pp. 67
Author(s):  
A. K. Mousa
Keyword(s):  
Residuated Lattice ◽  
Double Negation ◽  
Completely Distributive ◽  
Separation Axioms ◽  
Complete Residuated Lattice ◽  
Distributive Law ◽  
Double Negation Law ◽  
The Relationship

In this paper, we define and investigate the notions of \(L\)-separation axioms in \(L\)-fuzzifying supra-topology. Also, some of their characterizations and a systematic discussion on the relationship among these notions is gave in \(L\)-fuzzifying supra-topology where \(L\) is a complete residuated lattice. Sometimes we need more conditions on \(L\) such as the completely distributive law or that the "\(\wedge\)" is distributive over arbitrary joins or the double negation law as we illustrate through this paper. As applications of our work the corresponding results (see \cite{2, 13}) are generalized and new consequences are obtained.

scholarly journals Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 164
Author(s):  
Songsong Dai
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Distributive Law ◽  
Complete Orthomodular Lattice ◽  
The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

Semigroups Co-Ordinatizing Orthomodular Geometries

1965 ◽  
Vol 17 ◽  
pp. 40-51 ◽  
Author(s):  
D. J. Foulis
Keyword(s):  
Orthomodular Lattice ◽  
Orthomodular Lattices

In (2, 3, 4, and 5), the author has established a connection between orthomodular lattices and Baer *-semigroups. In brief, the connection is as follows. The lattice of closed projections of any Baer *-semigroup forms an orthomodular lattice. Conversely, if L is any orthomodular lattice, there exists a Baer *-semigroup S which co-ordinatizes L in the sense that L is isomorphic to the lattice of closed projections in S. In this note we shall assume that the reader is familiar with the results and the notation of the quoted papers.

Block-Finite Orthomodular Lattices

1979 ◽  
Vol 31 (5) ◽  
pp. 961-985 ◽  
Author(s):  
Günter Bruns
Keyword(s):  
Orthomodular Lattice ◽  
Boolean Algebras ◽  
Orthomodular Lattices ◽  
Special Situation ◽  
The Given ◽  
Insight Into ◽  
Considerable Insight

Introduction. Every orthomodular lattice (abbreviated : OML) is the union of its maximal Boolean subalgebras (blocks). The question thus arises how conversely Boolean algebras can be amalgamated in order to obtain an OML of which the given Boolean algebras are the blocks. This question we deal with in the present paper.The problem was first investigated by Greechie [6, 7, 8, 9]. His technique of pasting [6] will also play an important role in this paper. A case solved completely by Greechie [9] is the case that any two blocks intersect either in the bounds only or have the bounds, an atom and its complement in common. This is, of course, a very special situation. The more surprising it is that Greechie's methods, if skillfully applied, yield considerable insight into the structure of OMLs and provide a seemingly unexhaustible source for counter-examples.

On Bijective Correspondence between Fuzzy Reflexive Approximation Spaces and Fuzzy Transformation Systems

2020 ◽  
Vol 16 (02) ◽  
pp. 291-304
Author(s):  
Sutapa Mahato ◽  
S. P. Tiwari
Keyword(s):  
Lattice Structure ◽  
Approximation Space ◽  
Double Negation ◽  
Approximation Spaces ◽  
Bijective Correspondence ◽  
Fuzzy Approximation ◽  
Fuzzy Transformation ◽  
Double Negation Law ◽  
Transformation Systems ◽  
The Relationship

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law.

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