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

TL;DR

This paper introduces the BL_Chang logic, an extension of BL logic, and studies the algebraic structures formed by all ordinal sums of perfect MV-chains, linking them to Chang's MV-algebra.

Contribution

It defines the BL_Chang logic and characterizes the variety of algebras generated by all ordinal sums of perfect MV-chains, connecting it to Chang's MV-algebra.

Findings

BL_Chang logic is an axiomatic extension of BL.

The algebraic semantics of BL_Chang are analyzed in propositional and first-order cases.

The variety of BL_Chang-algebras is connected to the variety generated by Chang's MV-algebra.

Abstract

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.