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 Boolean sets, skew Boolean algebras and a non-commutative Stone duality

2015 ◽  
Vol 75 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Ganna Kudryavtseva ◽  
Mark V. Lawson
Keyword(s):  
Boolean Algebras ◽  
Stone Duality ◽  
Skew 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.

Skew Boolean algebras

1990 ◽  
Vol 27 (4) ◽  
pp. 497-506 ◽  
Author(s):  
Jonathan Leech
Keyword(s):  
Boolean Algebras ◽  
Skew Boolean Algebras

Skew Boolean algebras and discriminator varieties

1995 ◽  
Vol 33 (3) ◽  
pp. 387-398 ◽  
Author(s):  
R. J. Bignall ◽  
J. E. Leech
Keyword(s):  
Boolean Algebras ◽  
Discriminator Varieties ◽  
Skew Boolean Algebras

scholarly journals Free skew Boolean algebras

2016 ◽  
Vol 26 (07) ◽  
pp. 1323-1348 ◽  
Author(s):  
Ganna Kudryavtseva ◽  
Jonathan Leech
Keyword(s):  
Boolean Algebras ◽  
Free Algebras ◽  
Structure And Properties ◽  
Generating Sets ◽  
Central Elements ◽  
Skew Boolean Algebras

We study the structure and properties of free skew Boolean algebras (SBAs). For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements, and calculate the number of such elements. These results are used to study minimal generating sets of finite SBAs. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.

scholarly journals A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY

2013 ◽  
Vol 95 (3) ◽  
pp. 383-403 ◽  
Author(s):  
GANNA KUDRYAVTSEVA
Keyword(s):  
Boolean Algebras ◽  
Left Adjoint ◽  
Stone Duality ◽  
Twisted Product ◽  
Left Handed ◽  
Boolean Spaces ◽  
Skew Boolean Algebras ◽  
Noncommutative Setting

AbstractThe aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $.

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