We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category commutative is equivalent to MV-monoidal algebras. Roughly speaking, structures we call are without unary operation $x \mapsto -x$. The primitive operations $+$, $\lor$, $\land$, $0$, $1$, $-1$. A prime example these $\mathbb{R}$, with obvious interpretation operation...

Then it follows that mvm = m. Our main goal is to find an analogue of the Möbius transform on MV-algebras [4]. To this end, replace the boolean algebra 2X with a semisimple MV-algebra MX . There is no loss of generality in assuming that X is a compact Hausdorff space and MX is a separating MV-algebra of continuous functions over X. Since every Möbius transform mv corresponds to some finitely-ad...

MV-algebras were introduced by Chang, 1958 as algebraic bases for multi-valued logic. MV stands for "multi-valued" and MV algebras have already occupied an important place in the realm of nonstandard (mathematical) logic applied in several fields including cybernetics. In the present paper, using the Loomis-Sikorski theorem for cr-MV-algebras, we prove that, with every element a in a cr-MV alge...

The variety of MV-algebras is the equivalent algebraic semantics of the infinitely-valued Lukasiewicz logic [1], and MVn-algebras are algebraic models of the Lukasiewicz logic with n truth values (2 ≤ n < ω). Recall that an algebra A ∈ V is said to be a free algebra in a variety V, if there exists a set A0 ⊂ A such that A0 generates A and every mapping f from A0 to any algebra B ∈ V can be exte...

We consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $X$, $(X, forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $X$. We also show that, for a strong residuated lattice $X$, $bigcap {P_{lambda} ,|,P_{lambda} {rm is an} m{rm -filter} } = {1}$ and hence that any strong re...

In 2003, Di Nola, et.al. introduced the notion of MV -modules over a PMV algebra and A-ideals in MV -modules [5]. These are structures that naturally correspond to lu-modules over lu-rings [5]. Recall that an lu-ring is a pair (R,u), where (R, ⊕, ·, 0, ≤) is an l-ring and u is a strong unit of R (i.e, u is a strong unit of the underlying l-group) such that u · u ≤ u and l-ring is a structure (R...

Recently, the algebraic theory of MV -algebras is intensively studied. In this paper, we extend the concept of derivation of $MV$-algebras and we give someillustrative examples. Moreover, as a generalization of derivations of $MV$ -algebraswe introduce the notion of $f$-derivations and $(f; g)$-derivations of $MV$-algebras.Also, we investigate some properties of them.

This paper aims to give a clear and comprehensive view of the relations between the various classes of MV-algebras with product operations. The algebraic hyerarchy, from groups to algebras, can be transported by Γ-functors in order to provide the algebraic semantics for conservative extensions of Łukasiewicz logic. The MV-algebraic tensor product allows us to complete the picture with categoric...

States of unital Abelian lattice-groups (normalised positive group homomorphisms to R) provide an abstraction expected-value operators. A well-known theorem due Mundici asserts that the category (with unit-preserving lattice-group as morphisms) is equivalent algebraic MV-algebras, and their homomorphisms. Through this equivalence, states correspond certain [0,1]-valued functionals on which are ...

We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra P are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We pr...