{"title":"关于强次边值格的多样性","authors":"Sergio Celani, Hernán J. San Martín","doi":"10.1002/malq.202200067","DOIUrl":null,"url":null,"abstract":"<p>A subresiduated lattice is a pair <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>A</mi>\n <mo>,</mo>\n <mi>D</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(A,D)$</annotation>\n </semantics></math>, where <i>A</i> is a bounded distributive lattice, <i>D</i> is a bounded sublattice of <i>A</i> and for every <math>\n <semantics>\n <mrow>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n <mo>∈</mo>\n <mi>A</mi>\n </mrow>\n <annotation>$a,b\\in A$</annotation>\n </semantics></math> there exists the maximum of the set <math>\n <semantics>\n <mrow>\n <mo>{</mo>\n <mi>d</mi>\n <mo>∈</mo>\n <mi>D</mi>\n <mo>:</mo>\n <mi>a</mi>\n <mo>∧</mo>\n <mi>d</mi>\n <mo>≤</mo>\n <mi>b</mi>\n <mo>}</mo>\n </mrow>\n <annotation>$\\lbrace d\\in D:a\\wedge d\\le b\\rbrace$</annotation>\n </semantics></math>, which is denoted by <math>\n <semantics>\n <mrow>\n <mi>a</mi>\n <mo>→</mo>\n <mi>b</mi>\n </mrow>\n <annotation>$a\\rightarrow b$</annotation>\n </semantics></math>. This pair can be regarded as an algebra <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>A</mi>\n <mo>,</mo>\n <mo>∧</mo>\n <mo>,</mo>\n <mo>∨</mo>\n <mo>,</mo>\n <mo>→</mo>\n <mo>,</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$(A,\\wedge ,\\vee ,\\rightarrow ,0,1)$</annotation>\n </semantics></math> of type (2, 2, 2, 0, 0), where <math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <mo>=</mo>\n <mo>{</mo>\n <mi>a</mi>\n <mo>∈</mo>\n <mi>A</mi>\n <mo>:</mo>\n <mn>1</mn>\n <mo>→</mo>\n <mi>a</mi>\n <mo>=</mo>\n <mi>a</mi>\n <mo>}</mo>\n </mrow>\n <annotation>$D=\\lbrace a\\in A: 1\\rightarrow a =a\\rbrace$</annotation>\n </semantics></math>. The class of subresiduated lattices is a variety which properly contains the variety of Heyting algebras. In this paper we study the subvariety of subresiduated lattices, denoted by <math>\n <semantics>\n <msup>\n <mi>S</mi>\n <mo>□</mo>\n </msup>\n <annotation>$\\mathrm{S}^{\\Box }$</annotation>\n </semantics></math>, whose members satisfy the equation <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>→</mo>\n <mo>(</mo>\n <mi>a</mi>\n <mo>∨</mo>\n <mi>b</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>→</mo>\n <mi>a</mi>\n <mo>)</mo>\n <mo>∨</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>→</mo>\n <mi>b</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$1\\rightarrow (a\\vee b) = (1\\rightarrow a) \\vee (1\\rightarrow b)$</annotation>\n </semantics></math>. Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition <math>\n <semantics>\n <mrow>\n <mi>a</mi>\n <mo>→</mo>\n <mi>b</mi>\n <mo>∈</mo>\n <mo>{</mo>\n <mn>1</mn>\n <mo>→</mo>\n <mi>b</mi>\n <mo>,</mo>\n <mn>1</mn>\n <mo>}</mo>\n </mrow>\n <annotation>$a\\rightarrow b \\in \\lbrace 1\\rightarrow b,1\\rbrace$</annotation>\n </semantics></math> for every <math>\n <semantics>\n <mrow>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n </mrow>\n <annotation>$a,b$</annotation>\n </semantics></math> are satisfied, we also study the subvariety of <math>\n <semantics>\n <msup>\n <mi>S</mi>\n <mo>□</mo>\n </msup>\n <annotation>$\\mathrm{S}^{\\Box }$</annotation>\n </semantics></math> generated by the class whose members satisfy that <math>\n <semantics>\n <mrow>\n <mi>a</mi>\n <mo>→</mo>\n <mi>b</mi>\n <mo>∈</mo>\n <mo>{</mo>\n <mn>1</mn>\n <mo>→</mo>\n <mi>b</mi>\n <mo>,</mo>\n <mn>1</mn>\n <mo>}</mo>\n </mrow>\n <annotation>$a\\rightarrow b \\in \\lbrace 1\\rightarrow b,1\\rbrace$</annotation>\n </semantics></math> for every <math>\n <semantics>\n <mrow>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n </mrow>\n <annotation>$a,b$</annotation>\n </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 2","pages":"207-220"},"PeriodicalIF":0.4000,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the variety of strong subresiduated lattices\",\"authors\":\"Sergio Celani, Hernán J. San Martín\",\"doi\":\"10.1002/malq.202200067\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A subresiduated lattice is a pair <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>A</mi>\\n <mo>,</mo>\\n <mi>D</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(A,D)$</annotation>\\n </semantics></math>, where <i>A</i> is a bounded distributive lattice, <i>D</i> is a bounded sublattice of <i>A</i> and for every <math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n <mo>,</mo>\\n <mi>b</mi>\\n <mo>∈</mo>\\n <mi>A</mi>\\n </mrow>\\n <annotation>$a,b\\\\in A$</annotation>\\n </semantics></math> there exists the maximum of the set <math>\\n <semantics>\\n <mrow>\\n <mo>{</mo>\\n <mi>d</mi>\\n <mo>∈</mo>\\n <mi>D</mi>\\n <mo>:</mo>\\n <mi>a</mi>\\n <mo>∧</mo>\\n <mi>d</mi>\\n <mo>≤</mo>\\n <mi>b</mi>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$\\\\lbrace d\\\\in D:a\\\\wedge d\\\\le b\\\\rbrace$</annotation>\\n </semantics></math>, which is denoted by <math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n <mo>→</mo>\\n <mi>b</mi>\\n </mrow>\\n <annotation>$a\\\\rightarrow b$</annotation>\\n </semantics></math>. This pair can be regarded as an algebra <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>A</mi>\\n <mo>,</mo>\\n <mo>∧</mo>\\n <mo>,</mo>\\n <mo>∨</mo>\\n <mo>,</mo>\\n <mo>→</mo>\\n <mo>,</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(A,\\\\wedge ,\\\\vee ,\\\\rightarrow ,0,1)$</annotation>\\n </semantics></math> of type (2, 2, 2, 0, 0), where <math>\\n <semantics>\\n <mrow>\\n <mi>D</mi>\\n <mo>=</mo>\\n <mo>{</mo>\\n <mi>a</mi>\\n <mo>∈</mo>\\n <mi>A</mi>\\n <mo>:</mo>\\n <mn>1</mn>\\n <mo>→</mo>\\n <mi>a</mi>\\n <mo>=</mo>\\n <mi>a</mi>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$D=\\\\lbrace a\\\\in A: 1\\\\rightarrow a =a\\\\rbrace$</annotation>\\n </semantics></math>. The class of subresiduated lattices is a variety which properly contains the variety of Heyting algebras. In this paper we study the subvariety of subresiduated lattices, denoted by <math>\\n <semantics>\\n <msup>\\n <mi>S</mi>\\n <mo>□</mo>\\n </msup>\\n <annotation>$\\\\mathrm{S}^{\\\\Box }$</annotation>\\n </semantics></math>, whose members satisfy the equation <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>→</mo>\\n <mo>(</mo>\\n <mi>a</mi>\\n <mo>∨</mo>\\n <mi>b</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>→</mo>\\n <mi>a</mi>\\n <mo>)</mo>\\n <mo>∨</mo>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>→</mo>\\n <mi>b</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$1\\\\rightarrow (a\\\\vee b) = (1\\\\rightarrow a) \\\\vee (1\\\\rightarrow b)$</annotation>\\n </semantics></math>. Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition <math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n <mo>→</mo>\\n <mi>b</mi>\\n <mo>∈</mo>\\n <mo>{</mo>\\n <mn>1</mn>\\n <mo>→</mo>\\n <mi>b</mi>\\n <mo>,</mo>\\n <mn>1</mn>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$a\\\\rightarrow b \\\\in \\\\lbrace 1\\\\rightarrow b,1\\\\rbrace$</annotation>\\n </semantics></math> for every <math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n <mo>,</mo>\\n <mi>b</mi>\\n </mrow>\\n <annotation>$a,b$</annotation>\\n </semantics></math> are satisfied, we also study the subvariety of <math>\\n <semantics>\\n <msup>\\n <mi>S</mi>\\n <mo>□</mo>\\n </msup>\\n <annotation>$\\\\mathrm{S}^{\\\\Box }$</annotation>\\n </semantics></math> generated by the class whose members satisfy that <math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n <mo>→</mo>\\n <mi>b</mi>\\n <mo>∈</mo>\\n <mo>{</mo>\\n <mn>1</mn>\\n <mo>→</mo>\\n <mi>b</mi>\\n <mo>,</mo>\\n <mn>1</mn>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$a\\\\rightarrow b \\\\in \\\\lbrace 1\\\\rightarrow b,1\\\\rbrace$</annotation>\\n </semantics></math> for every <math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n <mo>,</mo>\\n <mi>b</mi>\\n </mrow>\\n <annotation>$a,b$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49864,\"journal\":{\"name\":\"Mathematical Logic Quarterly\",\"volume\":\"69 2\",\"pages\":\"207-220\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-07-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Logic Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200067\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200067","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
A subresiduated lattice is a pair , where A is a bounded distributive lattice, D is a bounded sublattice of A and for every there exists the maximum of the set , which is denoted by . This pair can be regarded as an algebra of type (2, 2, 2, 0, 0), where . The class of subresiduated lattices is a variety which properly contains the variety of Heyting algebras. In this paper we study the subvariety of subresiduated lattices, denoted by , whose members satisfy the equation . Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition for every are satisfied, we also study the subvariety of generated by the class whose members satisfy that for every .
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Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.
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