1,048 research outputs found
On some properties of quasi-MV algebras and square root quasi-MV algebras, IV
In the present paper, which is a sequel to
[20, 4, 12], we investigate further the structure theory of quasiMV
algebras and √0quasi-MV algebras. In particular: we provide
a new representation of arbitrary √0qMV algebras in terms
of √0qMV algebras arising out of their MV* term subreducts of
regular elements; we investigate in greater detail the structure
of the lattice of √0qMV varieties, proving that it is uncountable,
providing equational bases for some of its members, as well as
analysing a number of slices of special interest; we show that the
variety of √0qMV algebras has the amalgamation property; we
provide an axiomatisation of the 1-assertional logic of √0qMV
algebras; lastly, we reconsider the correspondence between Cartesian
√0qMV algebras and a category of Abelian lattice-ordered
groups with operators first addressed in [10]
The Logic of Quasi-MV Algebras
The algebraic theory of quasi-MV algebras, generalizations of MV algebras arising in quantum computation, is by now rather well-developed. Although it is possible to define several interesting logics from these structures, so far this aspect has not been investigated. The present article aims at filling this ga
Semicanonical bases and preprojective algebras
We study the multiplicative properties of the dual of Lusztig's semicanonical
basis.The elements of this basis are naturally indexed by theirreducible
components of Lusztig's nilpotent varieties, whichcan be interpreted as
varieties of modules over preprojective algebras.We prove that the product of
two dual semicanonical basis vectorsis again a dual semicanonical basis vector
provided the closure ofthe direct sum of thecorresponding two irreducible
components is again an irreducible component.It follows that the semicanonical
basis and the canonical basiscoincide if and only if we are in Dynkin type
with .Finally, we provide a detailed study of the varieties of
modules over the preprojectivealgebra of type .We show that in this case
the multiplicative properties ofthe dual semicanonical basis are controlled by
the Ringel form of a certain tubular algebra of type (6,3,2) and by
thecorresponding elliptic root system of type .Comment: Minor corrections. Final version to appear in Annales Scientifiques
de l'EN
Double affine Hecke algebras and affine flag manifolds, I
This lecture reviews the classification of simple modules of double affine
Hecke algebras via the K-theory of Steinberg varieties of affine typeComment: 52 page
On some Properties of quasi-MV √ Algebras and quasi-MV Algebras. Part IV
In the present paper, which is a sequel to [20, 4, 12], we investigate further the structure theory of quasi-MV algebras and √ quasi-MV algebras. In particular: we provide a new representation of arbitrary √ qMV algebras in terms of √ qMV algebras arising out of their MV* term subreducts of regular elements; we investigate in greater detail the structure of the lattice of √ qMV varieties, proving that it is uncountable, providing equational bases for some of its members, as well as analysing a number of slices of special interest; we show that the variety of √ qMV algebras has the amalgamation property; we provide an axiomatisation of the 1-assertional logic of √ qMV algebras; lastly, we reconsider the correspondence between Cartesian √ qMV algebras and a category of Abelian lattice-ordered groups with operators first addressed in [10]
Logical and algebraic structures from Quantum Computation
The main motivation for this thesis is given by the open problems regarding the axiomatisation of quantum computational logics. This thesis will be structured as follows: in Chapter 2 we will review some basics of universal algebra and functional analysis. In Chapters 3 through 6 the fundamentals of quantum gate theory will be produced. In Chapter 7 we will introduce quasi-MV algebras, a formal study of a suitable selection of algebraic operations associated with quantum gates. In Chapter 8 quasi-MV algebras will be expanded by a unary operation hereby dubbed square root of the inverse, formalising a quantum gate which allows to induce entanglement states. In Chapter 9 we will investigate some categorial dualities for the classes of algebras introduced in Chapters 7 and 8. In Chapter 10 the discriminator variety of linear Heyting quantum computational structures, an algebraic counterpart of the strong quantum computational logic, will be considered. In Chapter 11, we will list some open problems and, at the same time, draw some tentative conclusions. Lastly, in Chapter 12 we will provide a few examples of the previously investigated structures
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