{"title":"Some Properties of Hyperoperations and Hyperclones","authors":"Hajime Machida","doi":"10.1142/9789812704979_0022","DOIUrl":"https://doi.org/10.1142/9789812704979_0022","url":null,"abstract":"","PeriodicalId":265391,"journal":{"name":"Words, Languages & Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128943939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Semidirect Products With the Pseudovariety of All Finite Groups","authors":"J. Almeida, A. Escada","doi":"10.1142/9789812704979_0001","DOIUrl":"https://doi.org/10.1142/9789812704979_0001","url":null,"abstract":"","PeriodicalId":265391,"journal":{"name":"Words, Languages & Combinatorics","volume":"339 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124776134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Words Guaranteeing Minimal Image","authors":"S. Margolis, J. Pin, Mikhail Volkov","doi":"10.1142/9789812704979_0023","DOIUrl":"https://doi.org/10.1142/9789812704979_0023","url":null,"abstract":"Given a positive integer n and a finite alphabet Σ, a word w over Σ is said to guarantee minimum image if, for every homomorphism φ from the free monoid Σ∗ over Σ into the monoid of all transformations of an n-element set, the range of the transformation wφ has the minimum cardinality among the ranges of all transformations of the form vφ where v runs over Σ∗ . Although the existence of words guaranteeing minimum image is pretty obvious, the problem of their explicit description is very far from being trivial. Sauer and Stone in 1991 gave a recursive construction for such a word w but the length of their word was doubly exponential (as a function of n). We first show that some known results of automata theory immediately lead to an alternative construction that yields a simpler word that guarantees minimum image: it has exponential length, more precisely, its length is O(|Σ| 1 6 (n 3 ). Then with some more effort, we find a word guaranteeing minimum image similar to that of Sauer and Stone but of length O(|Σ| 1 2 (n 2 ). On the other hand, we prove that the length of any word guaranteeing minimum image cannot be less than |Σ| .","PeriodicalId":265391,"journal":{"name":"Words, Languages & Combinatorics","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124273078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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