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On some properties of quasi-MV algebras and √' quasi-MV algebras. Part IV

Jipsen P., Ledda A., Paoli F.

Reports on Mathematical Logic

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2013

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Vol. 48

3--36

EN

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 pro- vide a new representation of arbitrary√' MV algebras in terms of √'MV algebras arising out of their MV* term subreducts of regular elements; we investigate in greater detail the structure of the lattice of √' MV 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 √'MV algebras has the amalgamation property; we provide an axiomatisation of the 1-assertional logic of √'MV algebras; lastly, we reconsider the correspondence between Carte- sian √'MV algebras and a category of Abelian lattice-ordered groups with operators first addressed in [10].

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On some properties of quasi MV algebras and square root quasi MV algebras. Part III

Paoli F., Kowalski T.

Reports on Mathematical Logic

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2010

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Vol. 45

161-199

EN

In the present paper, which is a sequel to [14] and [3], we investigate further the structure theory of quasi-MV algebras and square root' quasi-MV algebras. In particular: we provide an improved version of the subdirect representation theorem for both varieties; we characterise the Ursini ideals of quasi-MV algebras; we establish a restricted version of J�Lonsson�fs lemma, again for both varieties; we simplify the proof of standard completeness for the variety of square root ' quasi-MV algebras; we show that this same va- riety has the finite embeddability property; finally, we investigate the structure of the lattice of subvarieties of Square root' quasi-MV algebras.

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On some properties of quasi-MV algebras and square root ' quasi-MV algebras

Paoli F., Ledda A., Freytes H.

Reports on Mathematical Logic

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2009

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Vol. 44

31-63

EN

We investigate some properties of two varieties of algebras arising from quantum computation - quasi-MV alge- bras and square root ' quasi-MV algebras - first introduced in [13], [12] and tightly connected with fuzzy logic. We establish the finite model property and the congruence extension property for both varieties; we characterize the quasi-MV reducts and subreducts of square root ' quasi- MV algebras; we give a representation of semisimple square root ' quasi-MV algebras in terms of algebras of functions; finally, we describe the structure of free algebras with one generator in both varieties.

4

Simplified affine phase structures

Paoli F.

Reports on Mathematical Logic

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1998

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No. 32

21-34

EN

Phase models for affine linear logic were independently devised by Lafont [10] and Piazza [15], although foreshadowed by Ono [14]. However, the existing semantics either contain no explicit directions for the construction of models in the general case, or else are forced to resort to additional conditions extending Girard's semantics (preordering of the monoids, the condition that the set of antiphases be an ideal). We dispense with these extra postulates - at least for the subexponential fragment of this logic - considering stuctures where the set of antiphases is concretely constructed. Moreover, we show the equivalence of our phase models and the algebraic models of affine linear logic by means of a representation theorem.