Skew Boolean algebras derived from generalized Boolean algebras

2008 ◽  
Vol 58 (3) ◽  
pp. 287-302 ◽  
Author(s):  
Jonathan Leech ◽  
Matthew Spinks
Keyword(s):  
Boolean Algebras ◽  
Skew Boolean Algebras ◽  
Generalized 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 Wreath Product Action on Generalized Boolean Algebras

10.37236/4831 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ashish Mishra ◽  
Murali K. Srinivasan
Keyword(s):  
Finite Group ◽  
Boolean Algebra ◽  
Wreath Product ◽  
Boolean Algebras ◽  
Product Action ◽  
Finite Set ◽  
A Finite Group ◽  
Multiplicity Free ◽  
Generalized Boolean Algebra ◽  
Generalized Boolean Algebras

Let $G$ be a finite group acting on the finite set $X$ such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product $G\sim S_n$ on the generalized Boolean algebra $B_X(n)$. We explicitly block diagonalize the commutant of this action.

On EMV-Semirings

2019 ◽  
Vol 69 (4) ◽  
pp. 739-752 ◽  
Author(s):  
R. A. Borzooei ◽  
M. Shenavaei ◽  
A. Di Nola ◽  
O. Zahiri
Keyword(s):  
Distributive Lattice ◽  
Algebraic Structure ◽  
Maximal Ideal ◽  
Boolean Algebras ◽  
Algebraic Extension ◽  
Theoretic Approach ◽  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Generalized Boolean Algebras

Abstract The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

Torsion classes of generalized Boolean algebras

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Analogous Result ◽  
Boolean Algebras ◽  
Radical Class ◽  
Torsion Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Brouwerian Lattice ◽  
Generalized Boolean Algebras

AbstractTorsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T g and R g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T g and R g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized 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

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

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.

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