{"title":"Heyting $$\\kappa $$-Frames","authors":"Hector Freytes, Giuseppe Sergioli","doi":"10.1007/s11225-023-10072-3","DOIUrl":null,"url":null,"abstract":"Abstract In the framework of algebras with infinitary operations, the equational theory of $$\\bigvee _{\\kappa }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mo>⋁</mml:mo> <mml:mi>κ</mml:mi> </mml:msub> </mml:math> -complete Heyting algebras or Heyting $$\\kappa $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>κ</mml:mi> </mml:math> -frames is studied. A Hilbert style calculus algebraizable in this class is formulated. Based on the infinitary structure of Heyting $$\\kappa $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>κ</mml:mi> </mml:math> -frames, an equational type completeness theorem related to the $$\\langle \\bigvee , \\wedge , \\rightarrow , 0 \\rangle $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>⋁</mml:mo> <mml:mo>,</mml:mo> <mml:mo>∧</mml:mo> <mml:mo>,</mml:mo> <mml:mo>→</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> -structure of frames is also obtained.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11225-023-10072-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
Abstract In the framework of algebras with infinitary operations, the equational theory of $$\bigvee _{\kappa }$$ ⋁κ -complete Heyting algebras or Heyting $$\kappa $$ κ -frames is studied. A Hilbert style calculus algebraizable in this class is formulated. Based on the infinitary structure of Heyting $$\kappa $$ κ -frames, an equational type completeness theorem related to the $$\langle \bigvee , \wedge , \rightarrow , 0 \rangle $$ ⟨⋁,∧,→,0⟩ -structure of frames is also obtained.
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