Proof. Suppose Mv = λv. We want to show that λ has imaginary value 0. For a complex number x = a + ib, the conjugate of x, is defined as follows: x∗ = a − ib. So, all we need to show is that λ = λ∗. The conjugate of a vector is the conjugate of all of its coordinate. Taking the conjugate transpose of both sides of the above equality, we have v∗M = λ∗v∗, (8.1) where we used that M = M . So, on o...

In this note we prove that it is impossible to extend the natural equivalence between the category of MV-algebras and the category of Abelian `-groups with strong unit described by C. C. Chang, 1958, and Cignoli & Mundici, 1997, to a natural equivalence between the category of m-zeroids and some category containing the category of Abelian `-groups with strong unit.

We prove that RS-BL-algebras are MV-algebras.

In this paper, for any non-empty subset C of a pseudo BCI-algebra X, the concept of p-closure of C, denoted by C(pc), is introduced and some related properties are investigated. Applying this concept, a characterization of the minimal elements of X is given. It is proved that C(pc) is the least closed pseudo BCI-ideal of X containing C and K(X) for any ideal C of X...

Pseudo-BL algebras are noncommutative generalizations of BL-algebras and they include pseudo-MV algebras, a class of structures that are categorically equivalent to l-groups with strong unit. In this paper we characterize directly indecomposable pseudo-BL algebras and we define and study different classes of these structures: local, good, perfect, peculiar, and (strongly) bipartite pseudo-BL al...

Coalition games are generalized to semisimple MV-algebras. Coalitions are viewed as [0,1]-valued functions on a set of players, which enables to express a degree of membership of a player in a coalition. Every game is a real-valued mapping on a semisimple MV-algebra. The goal is to recover the so-called core: a set of final distributions of payoffs, which are represented by measures on the MV-a...

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism φ, we assign an AF -algebra Aφ (an operator algebra). It is proved that the assignment is functorial, i.e. every φ, conjugate to φ, maps to an AF algebra Aφ′ , which is stably isomorphic to Aφ. The new invariants of the conjugacy of the pseudo-Anosov automorphisms are ob...

Let $f : A rightarrow B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we give a necessary and sufficient condition for the amalgamated algebra along an ideal $Abowtie^fJ$ to be $J$-Noetherian. Then we give a characterization for pseudo-irreducible ideals of $Abowtie^fJ$, in special cases.

Lukasiewicz implication algebras are the {→, 1}-subreducts of MV algebras. They are the algebraic counterpart of SuperLukasiewicz Implicational Logics. We give a description of free Lukasiewicz implication algebras in the context of McNaughton functions. More precisely, we show that the |X|-free Lukasiewicz implication algebra is isomorphic to ⋃ x∈X [xθ) for a certain congruence θ over the |X|-...

In this paper, by considering the notion of upsets we construct some topologies on BLalgebras. We show that any BL-algebras with respect to these topologies formes semitopological BLalgebra. Then we obtain some of the topological aspects of theses structures such as connectivity and compactness. Moreover, we introduced two kinds of semitopological MV -algebra by using two kinds of definition of...