{"title":"等价于多重代数范畴的有序代数范畴","authors":"M. Coniglio, Guilherme V. Toledo","doi":"10.18778/0138-0680.2023.23","DOIUrl":null,"url":null,"abstract":"It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\\(\\textit{CABA}\\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \\(\\textbf{Set}\\) and the category of \\(\\textit{CABA}\\)s.\nWe modify this result by taking multialgebras over a signature \\(\\Sigma\\), specifically those whose non-deterministic operations cannot return the empty-set, to \\(\\textit{CABA}\\)s with their zero element removed (which we call a \\({\\em bottomless Boolean algebra}\\)) equipped with a structure of \\(\\Sigma\\)-algebra compatible with its order (that we call \\({\\em ord-algebras}\\)). Conversely, an ord-algebra over \\(\\Sigma\\) is taken to its set of atomic elements equipped with a structure of multialgebra over \\(\\Sigma\\). This leads to an equivalence between the category of \\(\\Sigma\\)-multialgebras and the category of ord-algebras over \\(\\Sigma\\).\nThe intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Category of Ordered Algebras Equivalent to the Category of Multialgebras\",\"authors\":\"M. Coniglio, Guilherme V. Toledo\",\"doi\":\"10.18778/0138-0680.2023.23\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\\\\(\\\\textit{CABA}\\\\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \\\\(\\\\textbf{Set}\\\\) and the category of \\\\(\\\\textit{CABA}\\\\)s.\\nWe modify this result by taking multialgebras over a signature \\\\(\\\\Sigma\\\\), specifically those whose non-deterministic operations cannot return the empty-set, to \\\\(\\\\textit{CABA}\\\\)s with their zero element removed (which we call a \\\\({\\\\em bottomless Boolean algebra}\\\\)) equipped with a structure of \\\\(\\\\Sigma\\\\)-algebra compatible with its order (that we call \\\\({\\\\em ord-algebras}\\\\)). Conversely, an ord-algebra over \\\\(\\\\Sigma\\\\) is taken to its set of atomic elements equipped with a structure of multialgebra over \\\\(\\\\Sigma\\\\). This leads to an equivalence between the category of \\\\(\\\\Sigma\\\\)-multialgebras and the category of ord-algebras over \\\\(\\\\Sigma\\\\).\\nThe intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.\",\"PeriodicalId\":38667,\"journal\":{\"name\":\"Bulletin of the Section of Logic\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the Section of Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.18778/0138-0680.2023.23\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Arts and Humanities\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Section of Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.18778/0138-0680.2023.23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 0
摘要
众所周知,集合和完整的原子布尔代数(\(\textit{CABA}\)s)之间存在对应关系,将集合作为其幂集,反之,将完整的原子Boolean代数作为其原子元素集。当然,这种对应关系导致了\(\textbf{Set}\)的相反类别和\(\text it{CABA}\,到\(\textit{CABA}\)s,它们的零元素被移除(我们称之为\({\em无底布尔代数)),配备有与其阶兼容的\(\ Sigma\)-代数结构(我们称为\(\ em ord代数))。相反,在\(\西格玛\)上的一个ord代数被带到它的原子元素集,该原子元素集配备了在\(\西格玛\)之上的多代数的结构。这导致了\(\ Sigma\)-多代数的范畴和\(\西格玛\)上的ord代数的范畴之间的等价性。这里的直觉是,如果人们希望这样做,非决定论可能会被底层结构的足够丰富的排序所取代。
A Category of Ordered Algebras Equivalent to the Category of Multialgebras
It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s.
We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero element removed (which we call a \({\em bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \({\em ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of ord-algebras over \(\Sigma\).
The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.