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Logic"],"published-print":{"date-parts":[[2023,7]]},"abstract":"Abstract<\/jats:title>It is introduced a new algebra$$(A, \\otimes , \\oplus , *, \\rightharpoonup , 0, 1)$$<\/jats:tex-math>(<\/mml:mo>A<\/mml:mi>,<\/mml:mo>\u2297<\/mml:mo>,<\/mml:mo>\u2295<\/mml:mo>,<\/mml:mo>\u2217<\/mml:mo>,<\/mml:mo>\u21c0<\/mml:mo>,<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>called$$L_PG$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub>G<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-algebra if$$(A, \\otimes , \\oplus , *, 0, 1)$$<\/jats:tex-math>(<\/mml:mo>A<\/mml:mi>,<\/mml:mo>\u2297<\/mml:mo>,<\/mml:mo>\u2295<\/mml:mo>,<\/mml:mo>\u2217<\/mml:mo>,<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is$$L_P$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-algebra (i.e. an algebra from the variety generated by perfectMV<\/jats:italic>-algebras) and$$(A,\\rightharpoonup , 0, 1)$$<\/jats:tex-math>(<\/mml:mo>A<\/mml:mi>,<\/mml:mo>\u21c0<\/mml:mo>,<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is a G\u00f6del algebra (i.e. Heyting algebra satisfying the identity$$(x \\rightharpoonup y ) \\vee (y \\rightharpoonup x ) =1)$$<\/jats:tex-math>(<\/mml:mo>x<\/mml:mi>\u21c0<\/mml:mo>y<\/mml:mi>)<\/mml:mo>\u2228<\/mml:mo>(<\/mml:mo>y<\/mml:mi>\u21c0<\/mml:mo>x<\/mml:mi>)<\/mml:mo>=<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The lattice of congruences of an$$L_PG$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub>G<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-algebra$$(A, \\otimes , \\oplus , *, \\rightharpoonup , 0, 1)$$<\/jats:tex-math>(<\/mml:mo>A<\/mml:mi>,<\/mml:mo>\u2297<\/mml:mo>,<\/mml:mo>\u2295<\/mml:mo>,<\/mml:mo>\u2217<\/mml:mo>,<\/mml:mo>\u21c0<\/mml:mo>,<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is isomorphic to the lattice of Skolem filters (i.e. special type ofMV<\/jats:italic>-filters) of theMV<\/jats:italic>-algebra$$(A, \\otimes , \\oplus , *, 0, 1)$$<\/jats:tex-math>(<\/mml:mo>A<\/mml:mi>,<\/mml:mo>\u2297<\/mml:mo>,<\/mml:mo>\u2295<\/mml:mo>,<\/mml:mo>\u2217<\/mml:mo>,<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The variety$$\\mathbf {L_PG}$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub>G<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>of$$L_PG$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub>G<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-algebras is generated by the algebras$$(C, \\otimes , \\oplus , *, \\rightharpoonup , 0, 1)$$<\/jats:tex-math>(<\/mml:mo>C<\/mml:mi>,<\/mml:mo>\u2297<\/mml:mo>,<\/mml:mo>\u2295<\/mml:mo>,<\/mml:mo>\u2217<\/mml:mo>,<\/mml:mo>\u21c0<\/mml:mo>,<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>where$$(C, \\otimes , \\oplus , *, 0, 1)$$<\/jats:tex-math>(<\/mml:mo>C<\/mml:mi>,<\/mml:mo>\u2297<\/mml:mo>,<\/mml:mo>\u2295<\/mml:mo>,<\/mml:mo>\u2217<\/mml:mo>,<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is ChangMV<\/jats:italic>-algebra. Any$$L_PG$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub>G<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-algebra is bi-Heyting algebra. The set of theorems of the logic$$L_PG$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub>G<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is recursively enumerable. Moreover, we describe finitely generated free$$L_PG$$<\/jats:tex-math>L<\/mml:mi>P<\/mml:mi><\/mml:msub>G<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-algebras.<\/jats:p>","DOI":"10.1007\/s00153-023-00866-6","type":"journal-article","created":{"date-parts":[[2023,2,10]],"date-time":"2023-02-10T15:52:13Z","timestamp":1676044333000},"page":"789-809","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Involutive symmetric G\u00f6del spaces, their algebraic duals and logic"],"prefix":"10.1007","volume":"62","author":[{"given":"A.","family":"Di Nola","sequence":"first","affiliation":[]},{"given":"R.","family":"Grigolia","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5744-2334","authenticated-orcid":false,"given":"G.","family":"Vitale","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,10]]},"reference":[{"key":"866_CR1","doi-asserted-by":"publisher","first-page":"1356","DOI":"10.4153\/CJM-1986-069-0","volume":"38","author":"LP Belluce","year":"1986","unstructured":"Belluce, L.P.: Semisimple algebras of infinite-valued logic and bold fuzzy set theory. 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