A note on drastic product logic

TL;DR

This paper explores the properties and algebraic semantics of the drastic product logic (DP), a non-left-continuous t-norm based logic, establishing dualities, classifications, and the structure of its finite algebras.

Contribution

It justifies studying DP logic through algebraic properties, dualities, and classifications, despite its non-standard completeness and lack of residuum.

Findings

DP-algebras are dually equivalent to multisets of finite chains.

The paper classifies all axiomatic extensions of DP.

It computes free finitely generated DP-algebras.

Abstract

The drastic product $*_D$ is known to be the smallest $t$-norm, since $x *_D y = 0$ whenever $x, y < 1$. This $t$-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product $t$-norm based many-valued logics, in the sense of [EG01]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called ${\rm S}_{3}{\rm MTL}$ in [NOG06]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the $\Delta$ projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.