scholarly journals Commutative bounded integral residuated orthomodular lattices are Boolean algebras

2010 ◽  
Vol 15 (4) ◽  
pp. 635-636
Author(s):  
Josef Tkadlec ◽  
Esko Turunen
Keyword(s):  
Boolean Algebras ◽  
Orthomodular Lattices

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.

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.

Varieties of Orthomodular Lattices

1971 ◽  
Vol 23 (5) ◽  
pp. 802-810 ◽  
Author(s):  
Günter Bruns ◽  
Gudrun Kalmbach
Keyword(s):  
Principal Ideal ◽  
Boolean Algebras ◽  
Finitely Based ◽  
Orthomodular Lattices ◽  
Lattice Of Varieties ◽  
Equational Basis

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],

Varieties of Orthomodular Lattices. II

1972 ◽  
Vol 24 (2) ◽  
pp. 328-337 ◽  
Author(s):  
Günter Bruns ◽  
Gudrun Kalmbach
Keyword(s):  
Orthomodular Lattice ◽  
Oral Communication ◽  
Boolean Algebras ◽  
Vertical Line ◽  
Orthomodular Lattices ◽  
Lattice Of Varieties

In this paper we continue the study of equationally defined classes of orthomodular lattices started in [1].The only atom in the lattice of varieties of orthomodular lattices is the variety of all Boolean algebras. Every nontrivial variety contains it. It follows from B. Jónsson [4, Corollary 3.2] that the variety [MO2] generated by the orthomodular lattice MO2 of Figure 1 covers the variety of all Boolean algebras. I t was first shown by R. J. Greechie (oral communication) and is not difficult to see that every variety not consisting of Boolean algebras only contains [MO2]. Again it follows from the result of Jónsson's mentioned above that the varieties generated by one of the orthomodular lattices of Figures 2 to 5 cover [MO2]. The Figures 4 and 5 are to be understood in such a way that the orthocomplement of every element is on the vertical line through this element.

Central lifting property for orthomodular lattices

2020 ◽  
Vol 70 (6) ◽  
pp. 1307-1316
Author(s):  
Neda Arjomand Kermani ◽  
Esfandiar Eslami ◽  
Arsham Borumand Saeid
Keyword(s):  
Boolean Algebras ◽  
Prime Ideals ◽  
Direct Products ◽  
Orthomodular Lattices ◽  
Maximal Ideals ◽  
Central Elements ◽  
Simple Chain

AbstractWe introduce and investigate central lifting property (CLP) for orthomodular lattices as a property whereby all central elements can be lifted modulo every p-ideal. It is shown that prime ideals, maximal ideals and finite p-ideals have CLP. Also Boolean algebras, simple chain finite orthomodular lattices, subalgebras of an orthomodular lattices generated by two elements and finite orthomodular lattices have CLP. The main results of the present paper include the investigation of CLP for principal p-ideals and finite direct products of orthomodular lattices.

Basic algebras, logics, trends and applications

2015 ◽  
Vol 08 (03) ◽  
pp. 1550040 ◽  
Author(s):  
Ivan Chajda
Keyword(s):  
Quantum Mechanics ◽  
20Th Century ◽  
19Th Century ◽  
Classical Logic ◽  
Boolean Algebras ◽  
Heyting Algebras ◽  
Fuzzy Logics ◽  
Orthomodular Lattices ◽  
Mv Algebras ◽  
Intuitionistic Logics

The classical logic was axiomatized algebraically by means of Boolean algebras in 19th century by George Boole. Similar attempts went on 20th century for algebraic axiomatization of non-classical logics, e.g. intuitionistic logics (Brouwer and Heyting algebras), many-valued logics (Łukasiewicz, Chang’s MV-algebras, Post algebras), the logic of quantum mechanics (orthomodular lattices and posets) and fuzzy logics (residuated lattices). In this paper, we are focused in a common generalization of MV-algebras and orthomodular lattices. The resulting algebras, called basic algebras, have surprisingly strong and interesting properties and they can be investigated in their own. The aim of the paper is to get an overview of results reached during the last decade.

scholarly journals Sheffer operation in relational systems

2021 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Binary Relation ◽  
Boolean Algebras ◽  
Variety Of Algebras ◽  
Binary Relations ◽  
Orthomodular Lattices ◽  
Relational System ◽  
Distributive Variety ◽  
Congruence Distributive Variety ◽  
Relational Systems ◽  
Congruence Distributive

AbstractThe concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system, there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treatment of special groupoids which form a variety of algebras. If the Sheffer operation is also commutative, then the induced binary relation is antisymmetric. Moreover, commutative Sheffer groupoids form a congruence distributive variety. We characterize symmetry, antisymmetry and transitivity of binary relations by identities and quasi-identities satisfied by an assigned Sheffer operation. The concepts of twist products of relational systems and of Kleene relational systems are introduced. We prove that every directed relational system can be embedded into a directed relational system with involution via the twist product construction. If the relation in question is even transitive, then the directed relational system can be embedded into a Kleene relational system. Any Sheffer operation assigned to a directed relational system $${\mathbf {A}}$$ A with involution induces a Sheffer operation assigned to the twist product of $${\mathbf {A}}$$ A .

Basic semirings

2019 ◽  
Vol 69 (3) ◽  
pp. 533-540
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Classical Logic ◽  
Boolean Algebras ◽  
Basic Algebra ◽  
The Other ◽  
Natural Question ◽  
Orthomodular Lattices ◽  
Common Generalization ◽  
Boolean Rings ◽  
Mv Algebras

Abstract Basic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the references. Now we show that if a basic algebra is commutative then there exists a representation via certain semirings with involution similarly as it was done for MV-algebras by Belluce, Di Nola and Ferraioli. These so-called basic semirings, their ideals and congruences are studied in the paper.

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