关于强次边值格的多样性

Pub Date : 2023-07-11 DOI:10.1002/malq.202200067

Sergio Celani, Hernán J. San Martín

{"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":"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 A is a bounded distributive lattice, D is a bounded sublattice of A 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>.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":\"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 A is a bounded distributive lattice, D is a bounded sublattice of A 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>.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-07-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200067\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200067","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

子边值格是一对（A，D）$（A，D）$，其中A是有界分配格，D是A的有界子格，并且对于每个A，b∈A$A，b\在A$中存在集合{d∈d:A∧d≤b}$\l轨道d\在d:A楔d\le b\l轨道$中的最大值，其由→ b$a\右箭头b$。这对可以看作是一个代数（A，∧，→ , 0，1）$（A，\wedge，\vee，\rightarrow，0,1）$的类型（2，2，0，0），其中D={A∈A:1→ a=a｝$D=\l在a:1\rightarrowa=a\rbrace$中竞速a\。次边值格类是一个适当包含Heyting代数的变种的变种。在本文中，我们研究了由S表示的次边值格的子变种□ $\mathrm｛S｝^｛\Box｝$，其成员满足等式1→ （a∧b）=（1→ a）∧（1→ b）$1\rightarrow（a\vee b）=（1\rightarrowa）\vee（1\rigightarrowb）$。受以下事实的启发：在阶为全的任何次边值格中，前面的方程和条件a→ b∈{1→ b，1｝$a\rightarrow b\in\lblase 1\rightarrowb，1\rbrace$对于每个a，b$a，b$都满足，我们还研究了S的亚变种□ $\mathrm｛S｝^｛\Box｝$由其成员满足→ b∈{1→ b，1｝$a\rightarrowb\in\lblase 1\rightarrow b，1\rbrace$对于每个a，b$a，b$。

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On the variety of strong subresiduated lattices

A subresiduated lattice is a pair $(A, D)$ , where A is a bounded distributive lattice, D is a bounded sublattice of A and for every $a, b \in A$ there exists the maximum of the set ${d \in D : a \land d \leq b}$ , which is denoted by $a \to b$ . This pair can be regarded as an algebra $(A, \land, \lor, \to, 0, 1)$ of type (2, 2, 2, 0, 0), where $D = {a \in A : 1 \to a = a}$ . 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 $S^{□}$ , whose members satisfy the equation $1 \to (a \lor b) = (1 \to a) \lor (1 \to b)$ . Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition $a \to b \in {1 \to b, 1}$ for every $a, b$ are satisfied, we also study the subvariety of $S^{□}$ generated by the class whose members satisfy that $a \to b \in {1 \to b, 1}$ for every $a, b$ .

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