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

The connection of skew Boolean algebras and discriminator varieties to Church algebras

2015 ◽  
Vol 73 (3-4) ◽  
pp. 369-390 ◽  
Author(s):  
Karin Cvetko-Vah ◽  
Antonino Salibra
Keyword(s):  
Boolean Algebras ◽  
Discriminator Varieties ◽  
Skew Boolean Algebras

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

scholarly journals A refinement of Stone duality to skew Boolean algebras

2012 ◽  
Vol 67 (4) ◽  
pp. 397-416 ◽  
Author(s):  
Ganna Kudryavtseva
Keyword(s):  
Boolean Algebras ◽  
Stone Duality ◽  
Skew Boolean Algebras

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.

Skew Boolean algebras

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

scholarly journals Boolean like algebras

10.29007/dzwk ◽  
2018 ◽  
Author(s):  
Antonio Ledda ◽  
Tomasz Kowalski ◽  
Francesco Paoli ◽  
Antonino Salibra
Keyword(s):  
General Class ◽  
Central Element ◽  
Boolean Algebras ◽  
Similarity Type ◽  
Discriminator Varieties ◽  
Boolean Product ◽  
Exact Relationship

Using Vaggione's concept of central element in a double pointed algebra, weintroduce the notion of Boolean like variety as a generalisation ofBoolean algebras to an arbitrary similarity type. Appropriately relaxing therequirement that every element be central in any member of the variety, weobtain the more general class of semi-Boolean like varieties, whichstill retain many of the pleasing properties of Boolean algebras. We provethat a double pointed variety is discriminator iff it is semi-Boolean like,idempotent, and 0-regular. This theorem yields a new Maltsev-stylecharacterisation of double pointed discriminator varieties.Moreover, we point out the exact relationship between semi-Boolean-likevarieties and the quasi-discriminator varieties, and we providesemi-Boolean-like algebras with an explicit weak Boolean productrepresentation with directly indecomposable factors. Finally, we discuss idempotentsemi-Boolean-like algebras. We consider a noncommutative generalisation of Boolean algebras and prove - along the lines of similar results available for pointed discriminator varieties or for varieties with a commutative ternary deduction term that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional operations.

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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