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Register a new userBouligand-Severi Tangents in MV-Algebras
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In their recent seminal paper published in the Annals of Pure and Applied Logic, Dubuc and Poveda call an MV-algebra A strongly semisimple if all principal quotients of A are semisimple. All boolean algebras are strongly semisimple, and so are all finitely presented MV-algebras. We show that for any 1-generator MV-algebra semisimplicity is equivalent to strong semisimplicity. Further, a semisimple 2-generator MV-algebra A is strongly semisimple if and only if its maximal spectral space m(A) does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and m(A) has a Bouligand-Severi tangent then A is not strongly semisimple.
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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 provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.
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.
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 $mathcal A_{k,q}$.
A mistake concerning the ultra textit{LI}-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an textit{LI}-ideal to be an ultra textit{LI}-ideal are given. Moreover, the notion of an textit{LI}-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) textit{LI}-ideal and an textit{ILI}-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper textit{LI}-ideal and Boolean textit{LI}-ideal, (2) prime proper textit{LI}-ideal and textit{ILI}-ideal, (3) proper obstinate textit{LI}-ideal, (4) ultra textit{LI}-ideal.
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