Positive Subreducts in Finitely Generated Varieties of MV-algebras

Positive MV-algebras are negation-free and implication-free subreducts of MV-algebras. In this contribution we show that a finite axiomatic basis exists for the quasivariety of positive MV-algebras coming from any finitely generated variety of MV-algebras

Equivalence \`a la Mundici for lattice-ordered monoids

We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MMV-algebras (for `Monoidal MV-algebras'). Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation $x\mapsto -x$. The primitive operations are $+,\lor,\land,0,1,-1$. A prime example of these structures is $\mathbb{R}$, with the obvious interpretation of the operations. Analogously, MMV-algebras are MV-algebras without the negation $x\mapsto \lnot x$. The primitive operations are $\oplus,\odot,\lor,\land,0,1$. A motivating example of MMV-algebra is the negation-free reduct of the standard MV-algebra $[0,1]\subseteq \mathbb{R}$. We obtain the original Mundici's equivalence as a corollary of our main result

Square root of a multivector in 3D Clifford algebras

The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl3;0, Cl2;1, Cl1;2 and Cl0;3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.&nbsp

The variety generated by all the ordinal sums of perfect MV-chains

We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek - Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BL_Chang-algebras will be strictly connected to the one generated by Chang's MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some new results concerning these last structures and their logic.Comment: This is a revised version of the previous paper: the modifications concern essentially the presentation. The scientific content is substantially unchanged. The major variations are: Definition 2.7 has been improved. Section 3.1 has been made more compact. A new reference, [Bus04], has been added. There is some minor modification in Section 3.