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.

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],

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.

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.

States and Boolean Algebra

2008 ◽  
Vol 15 (04) ◽  
pp. 649-652
Author(s):  
Nabila N. Mikhaeel ◽  
Basim Samir Labib
Keyword(s):  
Mathematical Logic ◽  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Quantum Theories

We investigate subadditive measures on orthomodular lattices. We show as the main result that the Boolean algebra, the special metric orthomodular lattice and the orthomodular lattice which is unital with respect to subadditive states are equivalent. This result may find an application in the foundation of quantum theories and mathematical logic.

VARIETIES OF SKEW BOOLEAN ALGEBRAS WITH INTERSECTIONS

2016 ◽  
Vol 102 (2) ◽  
pp. 290-306
Author(s):  
JONATHAN LEECH ◽  
MATTHEW SPINKS
Keyword(s):  
Boolean Algebras ◽  
Lattice Of Varieties ◽  
Skew Boolean Algebras

Skew Boolean algebras for which pairs of elements have natural meets, called intersections, are studied from a universal algebraic perspective. Their lattice of varieties is described and shown to coincide with the lattice of quasi-varieties. Some connections of relevance to arbitrary skew Boolean algebras are also established.

scholarly journals Quantum categories for quantum logic

2019 ◽  
Vol 25 (1) ◽  
pp. 70-87
Author(s):  
Владимир Леонидович Васюков
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Heyting Algebra ◽  
Theoretic Approach ◽  
Orthomodular Lattices ◽  
C Algebras ◽  
Categorical Structure ◽  
Abstract Case ◽  
Complete Preorder ◽  
Categorical Semantics

The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors $\mathsf{[E,Sets]}$ where $\mathsf{E}$ is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called $quantos$ – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative $C^*$-algebras, which in turn are studied in their quantum habitat of noncommutative $C^*$-algebras – more fundamental structures than commutative $C^*$-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors $\mathsf{[E,QSets]}$ is constructed where $\mathsf{QSets}$ is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.

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