A Noncommutative Theory of Penrose Tilings

Mulvey, Christopher J.; Resende, Pedro

doi:10.1007/s10773-005-3997-2

Cited by 19 publications

(29 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where we have used stability of ς twice and the inequality follows from x ≤ ♦x and monotonicity of ς; this proves (10). Then (11) is proved in a similar way using the inequality x ≤ x.…”

Section: If ♦ and Are Conjugate We Havementioning

confidence: 70%

“…It remains to show that the pairs (α, α * ) and (α * , α) satisfy (10) and (11), which is done as follows:…”

Section: If ♦ and Are Conjugate We Havementioning

confidence: 99%

“…There are presentations of ssqs by generators and relations, which play the role of "Lindenbaum algebras" in the context of the quantalebased semantics. For instance, the "Lindenbaum quantale" for S5 is the pointed ssq Q S5 generated by the usual Lindenbaum algebra of S5 (with defining relations ensuring that it is a bounded sublattice of ς Q S5 ) with the distinguished point α ∈ Q S5 being subject to the relations α 2 = α * = α ≥ e. The usual Kripke structure based models for S5 can be identified with the relational representations of Q S5 , in other words the homomorphisms h : Q S5 → ℘( W × W ) of unital involutive quantales (see [10] for some properties of such representations). The same identification of models with the relational representations of a quantale applies to many other well known examples of modal logics.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An algebraic generalization of Kripke structures

Marcelino¹,

Resende²

2008

Math. Proc. Camb. Phil. Soc.

Self Cite

View full text Add to dashboard Cite

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL.

show abstract

Section: If ♦ and Are Conjugate We Havementioning

confidence: 70%

“…It remains to show that the pairs (α, α * ) and (α * , α) satisfy (10) and (11), which is done as follows:…”

Section: If ♦ and Are Conjugate We Havementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An algebraic generalization of Kripke structures

Marcelino¹,

Resende²

2008

Math. Proc. Camb. Phil. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…6 By framedistributivity, it is easy to show that, for any FA-quantale L, the atoms of L, being completely join-irreducible, are completely join-prime. 7 This implies for instance that .) is similar.…”

Section: Immediate From Item 1 and The Observation That At(l) ⊆ I(l)mentioning

confidence: 99%

Relational Representation of Groupoid Quantales

Palmigiano

2011

Order

View full text Add to dashboard Cite

“…For example, the spectrum of a C * -algebra or the "space" of Penrose tilings can be viewed as a quantale [24,30]. On the other hand, if the elements ∈ Q of a quantale Q are interpreted as logical propositions, the union A stands for the disjunction, while · features the sequential conjunction " and then ".…”

Section: Definition 13mentioning

confidence: 99%

Quantum B-algebras

Rump

2013

Open Mathematics

View full text Add to dashboard Cite

The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL-or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor "spectrum", it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete. MSC:03G12, 03G27, 06F07, 06F15, 06F35, 03F52, 03G25

show abstract

A Noncommutative Theory of Penrose Tilings

Cited by 19 publications

References 22 publications

An algebraic generalization of Kripke structures

An algebraic generalization of Kripke structures

Relational Representation of Groupoid Quantales

Quantum B-algebras

Contact Info

Product

Resources

About