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Register a new userHopfian $ell$-groups, MV-algebras and AF C$^*$-algebras
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Authors
Daniele Mundici
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An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself. We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and MV-algebras. Using Elliott classification and $K_0$-theory, we apply our results to other related structures, notably the Farey-Stern-Brocot AF C$^*$-algebra and all its primitive quotients, including the Behnke-Leptin C$^*$-algebras $mathcal A_{k,q}$.
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We consider algebras with basis numerated by elements of a group $G.$ We fix a function $f$ from $Gtimes G$ to a ground field and give a multiplication of the algebra which depends on $f$. We study the basic properties of such algebras. In particular, we find a condition on $f$ under which the corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup $hat G$ of $G$ we define a $hat G$-periodic algebra, which corresponds to a $hat G$-periodic function $f,$ we establish a criterion for the right nilpotency of a $hat G$-periodic algebra. In addition, for $G=mathbb Z$ we describe all $2mathbb Z$- and $3mathbb Z$-periodic algebras. Some properties of $nmathbb Z$-periodic algebras are obtained.
For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras.
For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise L ukasiewicz negation $ eg x={ eg alphamid alphain x}$, (truncated) Minkowski sum, $xoplus y={alphaoplus betamid alpha in x,,,betain y}$, pointwise L ukasiewicz conjunction $xodot y= eg( eg xoplus eg y)$, the operators $Delta x=[min x,min x]$, $ abla x=[max x,max x]$, and distinguished constants $0=[0,0],,, 1=[1,1],,,, mathsf{i} = A$. We list a few equations satisfied by the algebra $mathcal I(A)=(I(A),0,1,mathsf{i}, eg,Delta, abla,oplus,odot)$, call IMV-algebra every model ofthese equations, and show that, conversely, every IMV-algebra is isomorphic to the IMV-algebra $mathcal I(B)$ of all intervals in some MV-algebra $B$. We show that IMV-algebras are categorically equivalent to MV-algebras, and give a representation of free IMV-algebras. We construct L ukasiewicz interval logic, with its coNP-complete consequence relation, which we prove to be complete for $mathcal I([0,1])$-valuations. For any class $mathsf{Q}$ of partially ordered algebras with operations that are monotone or antimonotone in each variable, we consider the generalization $mathcal I_{mathsf{Q}}$ of the MV-algebraic functor $mathcal I$, and give necessary and sufficient conditions for $mathcal I_{mathsf{Q}}$ to be a categorical equivalence. These conditions are satisfied, e.g., by all subquasivarieties of residuated lattices.
We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras.
We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.
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