Logics of √’qMV algebras

Antonio Ledda

Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks

Università di Cagliari

Siena, September 8th 2008

Some motivation

qMV algebras were introduced in an attempt

to provide a convenient abstraction of the

algebra over the set of all density

operators of the two-dimensional complex

Hilbert space, endowed with a suitable

stock of quantum gates.

The definition of qMV-algebra

Definition

Łukasiewicz’s axiom Smoothness axioms

qMV-algebras

Adding the square root of the negation

√’qMV algebras were introduced as term

expansions of quasi-MV algebras by an

operation of square root of the

negation.

Adding the square root of the negation

quasi-Wajsberg algebras

Term equivalence

Theorem

The standard Wajsberg algebra St

The algebra F[0,1]

The standard qW algebras S and D

Equationally defined preorder

An example of equationally defined preorder

Logics from equationally preordered classes

Remark

A logic from an equationally preordered variety

The quasi-Łukasiewicz logic qŁ

A remark

Summary of the logic results

A “logical” version of qMV

Term equivalences

Logics of qMV algebras (1)

Logics of qMV algebras (2)

Most logics in the previous schema look noteworthy under some respect:

Logics of qMV algebras (3)

1-cartesian algebras

Examples

Inclusion relationships

Placing our logics in the Leibniz hierarchy (1)

Well-behaved logics is regularly algebraisable and is its

equivalent quasivariety semantics;

is regularly algebraisable and is its equivalent quasivariety semantics;

(they are the 1-assertional logics of relatively

1-regular quasivarieties)

Placing our logics in the Leibniz hierarchy (2)

Ill-behaved logicsNone of the other logics is protoalgebraic:

: the Leibniz operator is not monotone on the deductive filters of F120;

: it is a sublogic of such;

: the Leibniz operator is not monotone on the deductive filters of ;

Placing our logics in the Leibniz hierarchy (2)

Placing our logics in the Frege hierarchy

Selfextensional logics

Non-selfextensional logics

Some notations

We use the following abbreviations:

The logics C and C1

The logics C and C1

A completeness result

The notion of (strong) implicative filter

Remark

In the definition of strong implicative filter conditions F2, F3, F4, F5 are redundant

A characterization of the deductive filters

Thank you for your attention!!