Propositional skew Boolean logic

Bignall, Robert J; Spinks, Matthew

doi:10.1109/ismvl.1996.508334

Cited by 9 publications

(5 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(See [1,2,3,5,6,4] and [18,19].) For further background on skew Boolean algebras, see [1,8,12] and [14].…”

Section: Introductionmentioning

confidence: 99%

“…(2) Recall that a rectangular skew lattice is any skew lattice D = D; ∨, ∧ for which a ∧ b ∧ a = a = a ∨ b ∨ a holds for all a, b ∈ D. Rectangular skew lattices are also characterized by the identity x ∧ y ≈ y ∨ x. To within isomorphism, a rectangular skew lattice is given by defining ∨ and ∧ on a Cartesian product is given by setting | i ∈ I} of primitive algebras, S = i∈I P i is a skew Boolean algebra that is both complete and atomic.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Skew Boolean algebras derived from generalized Boolean algebras

Leech

Spinks

2008

Algebra univers.

View full text Add to dashboard Cite

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/D where D is the Green's relation defined initially on semigroups. In this paper we study skew Boolean algebras ω(B) constructed from generalized Boolean algebras B by a twisted product construction for which ω(B)/D ∼ = B. In particular we study the congruence lattice of ω(B) with an eye to viewing ω(B) as a minimal skew Boolean cover of B. This construction is the object part of a functor ω : GB → LSB from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor Ω : LSB → GB. BackgroundRecall first that a skew lattice is an algebra S = S; ∨, ∧ with associative, idempotent binary operations ∨ and ∧ that satisfy the absorption identities:Equivalently, ∨ and ∧ jointly satisfy the dualities a ∨ b = a iff a ∧ b = b and a ∨ b = b iff a ∧ b = a. Every skew lattice has a natural partial order given by aThe relation ≥ is a special case of commutativity. A skew lattice can have many instances of commutativity. In general these instances can be ambiguous in that only one of say a ∧ b = b ∧ a or a∨b = b∨a need hold. A skew lattice S is symmetric if a∧b = b∧a iff a∨b = b∨a for all a, b ∈ S, thus making commutativity unambiguous. A distinguished element 0 in a skew lattice S is a zero if 0 ∧ a = 0 = a ∧ 0, and thus 0 ∨ a = a = a ∨ 0, for all a ∈ S.A Boolean skew lattice is any symmetric skew lattice with a zero S; ∨, ∧, 0 such that for all a ∈ S, the principal poset ideal ⌈a⌉ = {b ∈ S | a ≥ b} is a Boolean sublattice of S. In this case, the relative complements \ on all these poset ideals

show abstract

“…(See [1,2,3,5,6,4] and [18,19].) For further background on skew Boolean algebras, see [1,8,12] and [14].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Skew Boolean algebras derived from generalized Boolean algebras

Leech

Spinks

2008

Algebra univers.

View full text Add to dashboard Cite

show abstract

“…Indeed, (2) follows from (1) by horizontal duality (new x ∨ y = old y ∨ x; new x ∧ y = old y ∧ x). As for (3), the equivalence of (i) with (ii) follows from the corresponding equivalences of (1) and (2). The equivalence of (ii) with (iii) follows from the corresponding equivalences of (1) and (2) and Corollary 4.7.…”

Section: Characterizing Cancellative Skew Latticesmentioning

confidence: 87%

Cancellation in Skew Lattices

Cvetko-Vah

Kinyon

Leech

et al. 2010

Order

View full text Add to dashboard Cite

Abstract. Distributive lattices are well known to be precisely those lattices that possess cancellation: x ∨ y = x ∨ z and x ∧ y = x ∧ z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the 5-element lattices M 3 or N 5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations ∧ and ∨ no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M 3 or N 5 that insure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x∨z = y∨z and x∧z = y∧z imply x = y) or just cancellative (satisfying both implications). We also present systems of identities showing that left [right or fully] cancellative skew lattices form varieties. Finally, we give some positive characterizations of cancellation.

show abstract

“…See [7] for general results on normal skew lattices. Their importance is due in part to skew Boolean algebras being normal as skew lattices [1,2,12,13,15]. Some nice counting theorems for categorical and strictly categorical skew lattices are given in [14].…”

Section: Forbidden Subalgebrasmentioning

confidence: 99%