Representation of Perfect and Local MV-algebras

Algebraic expansions of logics and algebras and a case study of Abelian l-groups and perfect MV-algebras

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \land p = q$. For a logic $L$ algebraized by a quasivariety $\mathcal{Q}$ we show that the AE-subclasses of $\mathcal{Q}$ correspond to certain natural expansions of $L$, which we call {\em algebraic expansions}. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of Abelian $\ell$-groups and perfect MV-algebras, thus fully describing the algebraic expansions of their …

风筝与伪 MV-代数的表示

Kites and representations of pseudo MV-algebras

本文研究了完美伪 MV-代数的结构，证明其可表示为风筝的广义版本且范畴等价于带特定自同构的 $\ell$-群，并刻画了由风筝生成的簇及其格结构。

Perfect extensions of de Morgan algebras

An algebra $\mathbb A$ is called a perfect extension of its subalgebra $\mathbb B$ if every congruence of $\mathbb B$ has a unique extension to $\mathbb A$. This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that $\mathbb A$ is a congruence-preserving extension of $\mathbb B$. Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra $\mathbb M$ is perfect extension of its Boolean subalgebra $B(\mat…

Congruence pairs of principal MS-algebras and perfect extensions

The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for $K_2$-algebras \cite{6}, is introduced. It is proved that the congruences of the principal MS-algebras $L$ correspond to the MS-congruence pairs on simpler substructures $L^{\circ\circ}$ and $D(L)$ of $L$ that were associated to~$L$ in \cite{4}. An analogy of a well-known Grätzer's problem \cite[Problem 57]{11} formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Pro…

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. Hájek - 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 las…

Perfect IFG-formulas

IFG logic is a variant of the independence-friendly logic of Hintikka and Sandu. We answer the question: ``Which IFG-formulas are equivalent to ordinary first-order formulas?'' We use the answer to show that the ordinary cylindric set algebra over a structure can be embedded into a reduct of the IFG-cylindric set algebra over the structure.