Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.

Rational invariants on the space of all structures of algebras on a two-dimensional vector space

Rational invariants on the space of all structures of algebras on a two-dimensional vector space

Orientation in operator algebras

A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics.

Weighted composition operators on the predual and dual Banach spaces of Beurling algebras on Z

Let vv be a weight sequence on ZZ and let ψ,φψ,φ be complex-valued functions on ZZ such that φ(Z)⊂Zφ(Z)⊂Z. In this paper we study the boundedness, compactness and weak compactness of weighted composition operators Cψ,φCψ,φ on predual Banach spaces c0(Z,1/v)c0(Z,1/v) and dual Banach spaces ℓ∞(Z,1/v)ℓ∞(Z,1/v) of Beurling algebras ℓ1(Z,v)ℓ1(Z,v).

Algebras and their arithmetics,

Reprint of the 1923 edition.

Linear algebras

Mode of access: Internet.

Arithmetica especulativa, y practica, y arte de algebra : en la qual se contiene todo lo que pertenece al arte menor, ò mercantil, y à las dos algebras, racional, è irracional, con la explicación de todas las proposiciones, y problemas de los libros quinto, septimo, octavo, nono y dezimo del principe de la matematica Euclides /

Mode of access: Internet.

Arithmetica especulativa y practiva [sic] y arte de algebra : en la qual se contiene todo lo qe pertenece al arte menor o mercantil, y a las dos algebras, racional e irracional : con la explicacion de todas las proposiciones y problemas de los libros quinto, septimo, octavo, nono y dezimo del principe de la mathematica Euclides /

Mode of access: Internet.

Arithmetica especulativa, y practica, y arte de algebra : en la qual se contiene todo lo que pertenece al arte menor, ò mercantivol, y à las dos algebras, racional, è irracional, con la explicación de todas las proposiciones, y problemas de los libros quinto, septimo, octavo, nono y decimo del principe de la matematica Euclides /

Mode of access: Internet.

Arithmetica especulativa, y practica, y arte de algebra : en la qual se contiene todo lo que pertenece al arte menor, ò mercantil, y à las dos algebras, racional, è irracional, con la explicación de todas las proposiciones, y problemas de los libros quinto, septimo, octavo, nono y dezimo del principe de la matematica Euclides /

Mode of access: Internet.

Arithmetica especulativa, y practica, y arte de algebra : en la qual se contiene todo lo que pertenece al arte menor, ò mercantil, y à las dos algebras, racional, è irracional, con la explicación de todas las proposiciones, y problemas de los libros quinto, septimo, octavo, nono y decimo del principe de la matematica Euclides /

Mode of access: Internet.

On the structure of Hopf algebras

Caption title.