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Mathematics 2019, 7(1), 107; https://doi.org/10.3390/math7010107

Fixpointed Idempotent Uninorm (Based) Logics

Department of Philosophy & Institute of Critical Thinking and Writing, Colleges of Humanities & Social Science Blvd., Chonbuk National University, Rm 417, Jeonju 54896, Korea
Received: 8 December 2018 / Revised: 14 January 2019 / Accepted: 18 January 2019 / Published: 20 January 2019
(This article belongs to the Special Issue Fuzziness and Mathematical Logic)
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Abstract

Idempotent uninorms are simply defined by fixpointed negations. These uninorms, called here fixpointed idempotent uninorms, have been extensively studied because of their simplicity, whereas logics characterizing such uninorms have not. Recently, fixpointed uninorm mingle logic (fUML) was introduced, and its standard completeness, i.e., completeness on real unit interval [ 0 , 1 ] , was proved by Baldi and Ciabattoni. However, their proof is not algebraic and does not shed any light on the algebraic feature by which an idempotent uninorm is characterized, using operations defined by a fixpointed negation. To shed a light on this feature, this paper algebraically investigates logics based on fixpointed idempotent uninorms. First, several such logics are introduced as axiomatic extensions of uninorm mingle logic (UML). The algebraic structures corresponding to the systems are then defined, and the results of the associated algebraic completeness are provided. Next, standard completeness is established for the systems using an Esteva–Godo-style approach for proving standard completeness. View Full-Text
Keywords: substructural fuzzy logic; algebraic completeness; standard completeness; fixpoint; idempotent uninorm substructural fuzzy logic; algebraic completeness; standard completeness; fixpoint; idempotent uninorm
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Yang, E. Fixpointed Idempotent Uninorm (Based) Logics. Mathematics 2019, 7, 107.

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