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 S_{3}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.

A note on drastic product logic / S. Aguzzoli, M. Bianchi, D. Valota (COMMUNICATIONS IN COMPUTER AND INFORMATION SCIENCE). - In: Information processing and management of uncertainty in knowledge-based systems : 15th International conference on information processing and management of uncertainty in knowledge-based systems, IPMU 2014 : Montpellier, France, july 15-19, 2014 : proceedings. Part 2 / [a cura di] A. Laurent, O. Strauss, B. Bouchon-Meunier, R.R. Yager. - Berlin : Springer, 2014. - ISBN 9783319088549. - pp. 365-374 (( Intervento presentato al 15. convegno International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU) tenutosi a Montpellier nel 2014 [10.1007/978-3-319-08855-6_37].

### A note on drastic product logic

#### 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 S_{3}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.
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Settore INF/01 - Informatica
2014
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/236997`
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