<abstract xmlns="http://www.w3.org/1999/xhtml">In the article we present in the Mizar system [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_001_w2aab3b7b1b1b6b1ab1ab1Aa">1</xref>], [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_008_w2aab3b7b1b1b6b1ab1ab8Aa">8</xref>] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_013_w2aab3b7b1b1b6b1ab1ac13Aa">13</xref>]. The name <italic>triangular</italic> emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_002_w2aab3b7b1b1b6b1ab1ab2Aa">2</xref>].After defining corresponding Mizar mode using four attributes, we introduced the following t-norms:
<list list-type="bullet"><list-item>minimum t-norm <monospace>minnorm</monospace> (Def. 6),</list-item><list-item>product t-norm <monospace>prodnorm</monospace> (Def. 8),</list-item><list-item>Łukasiewicz t-norm <monospace>Lukasiewicz_norm</monospace> (Def. 10),</list-item><list-item>drastic t-norm <monospace>drastic_norm</monospace> (Def. 11),</list-item><list-item>nilpotent minimum <monospace>nilmin_norm</monospace> (Def. 12),</list-item><list-item>Hamacher product <monospace>Hamacher_norm</monospace> (Def. 13),</list-item></list>
and corresponding t-conorms:
<list list-type="bullet"><list-item>maximum t-conorm <monospace>maxnorm</monospace> (Def. 7),</list-item><list-item>probabilistic sum <monospace>probsum_conorm</monospace> (Def. 9),</list-item><list-item>bounded sum <monospace>BoundedSum_conorm</monospace> (Def. 19),</list-item><list-item>drastic t-conorm <monospace>drastic_conorm</monospace> (Def. 14),</list-item><list-item>nilpotent maximum <monospace>nilmax_conorm</monospace> (Def. 18),</list-item><list-item>Hamacher t-conorm <monospace>Hamacher_conorm</monospace> (Def. 17).</list-item></list>Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_010_w2aab3b7b1b1b6b1ab1ac10Aa">10</xref>], [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_009_w2aab3b7b1b1b6b1ab1ab9Aa">9</xref>]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm).This work is a continuation of the development of fuzzy sets in Mizar [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_006_w2aab3b7b1b1b6b1ab1ab6Aa">6</xref>] started in [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_011_w2aab3b7b1b1b6b1ab1ac11Aa">11</xref>] and [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_003_w2aab3b7b1b1b6b1ab1ab3Aa">3</xref>]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_004_w2aab3b7b1b1b6b1ab1ab4Aa">4</xref>], the approach which was chosen allows however for merging both theories [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_005_w2aab3b7b1b1b6b1ab1ab5Aa">5</xref>], [<xref ref-type="bibr" rid="j_forma-2017-0009_ref_007_w2aab3b7b1b1b6b1ab1ab7Aa">7</xref>].</abstract>

In the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [2].After defining corresponding Mizar mode using four attributes, we introduced the following t-norms:
minimum t-norm minnorm (Def. 6),product t-norm prodnorm (Def. 8),Łukasiewicz t-norm Lukasiewicz_norm (Def. 10),drastic t-norm drastic_norm (Def. 11),nilpotent minimum nilmin_norm (Def. 12),Hamacher product Hamacher_norm (Def. 13),
and corresponding t-conorms:
maximum t-conorm maxnorm (Def. 7),probabilistic sum probsum_conorm (Def. 9),bounded sum BoundedSum_conorm (Def. 19),drastic t-conorm drastic_conorm (Def. 14),nilpotent maximum nilmax_conorm (Def. 18),Hamacher t-conorm Hamacher_conorm (Def. 17).Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [10], [9]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm).This work is a continuation of the development of fuzzy sets in Mizar [6] started in [11] and [3]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [4], the approach which was chosen allows however for merging both theories [5], [7].

Basic Formal Properties of Triangular Norms and Conorms

Formalized Mathematics

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

{"article-title":"Basic Formal Properties of Triangular Norms and Conorms"}

In the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The...

Basic Formal Properties of Triangular Norms and Conorms

Published Online: Sep 23, 2017

Page range: 93 - 100

Received: Jun 27, 2017

DOI: https://doi.org/10.1515/forma-2017-0009

Keywords
fuzzy set, triangular norm, triangular conorm, fuzzy logic

© 2017 Adam Grabowski, published by De Gruyter Open

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Basic Formal Properties of Triangular Norms and Conorms

Published Online: Sep 23, 2017

Page range: 93 - 100

Received: Jun 27, 2017

DOI: https://doi.org/10.1515/forma-2017-0009

Keywordsfuzzy set, triangular norm, triangular conorm, fuzzy logic

© 2017 Adam Grabowski, published by De Gruyter Open

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Keywords
fuzzy set, triangular norm, triangular conorm, fuzzy logic