Get quick answers to your questions about the article from our AI researcher chatbot
{'id': 'https://openalex.org/W3104722361', 'doi': 'https://doi.org/10.1007/s00233-016-9783-z', 'title': 'Ranks of finite semigroups of one-dimensional cellular automata', 'display_name': 'Ranks of finite semigroups of one-dimensional cellular automata', 'publication_year': 2016, 'publication_date': '2016-03-03', 'ids': {'openalex': 'https://openalex.org/W3104722361', 'doi': 'https://doi.org/10.1007/s00233-016-9783-z', 'mag': '3104722361'}, 'language': 'en', 'primary_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1007/s00233-016-9783-z', 'pdf_url': 'https://link.springer.com/content/pdf/10.1007%2Fs00233-016-9783-z.pdf', 'source': {'id': 'https://openalex.org/S75897683', 'display_name': 'Semigroup Forum', 'issn_l': '0037-1912', 'issn': ['0037-1912', '1432-2137'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310319900', 'host_organization_name': 'Springer Science+Business Media', 'host_organization_lineage': ['https://openalex.org/P4310319965', 'https://openalex.org/P4310319900'], 'host_organization_lineage_names': ['Springer Nature', 'Springer Science+Business Media'], 'type': 'journal'}, 'license': 'cc-by', 'license_id': 'https://openalex.org/licenses/cc-by', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'type': 'article', 'type_crossref': 'journal-article', 'indexed_in': ['arxiv', 'crossref', 'datacite'], 'open_access': {'is_oa': True, 'oa_status': 'hybrid', 'oa_url': 'https://link.springer.com/content/pdf/10.1007%2Fs00233-016-9783-z.pdf', 'any_repository_has_fulltext': True}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5035086666', 'display_name': 'Alonso Castillo-Ramirez', 'orcid': 'https://orcid.org/0000-0002-2020-8699'}, 'institutions': [{'id': 'https://openalex.org/I190082696', 'display_name': 'Durham University', 'ror': 'https://ror.org/01v29qb04', 'country_code': 'GB', 'type': 'education', 'lineage': ['https://openalex.org/I190082696']}], 'countries': ['GB'], 'is_corresponding': False, 'raw_author_name': 'Alonso Castillo-Ramirez', 'raw_affiliation_strings': ['School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK'], 'affiliations': [{'raw_affiliation_string': 'School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK', 'institution_ids': ['https://openalex.org/I190082696']}]}, {'author_position': 'last', 'author': {'id': 'https://openalex.org/A5037486046', 'display_name': 'Maximilien Gadouleau', 'orcid': 'https://orcid.org/0000-0003-4701-738X'}, 'institutions': [{'id': 'https://openalex.org/I190082696', 'display_name': 'Durham University', 'ror': 'https://ror.org/01v29qb04', 'country_code': 'GB', 'type': 'education', 'lineage': ['https://openalex.org/I190082696']}], 'countries': ['GB'], 'is_corresponding': False, 'raw_author_name': 'Maximilien Gadouleau', 'raw_affiliation_strings': ['School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK'], 'affiliations': [{'raw_affiliation_string': 'School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK', 'institution_ids': ['https://openalex.org/I190082696']}]}], 'institution_assertions': [], 'countries_distinct_count': 1, 'institutions_distinct_count': 1, 'corresponding_author_ids': [], 'corresponding_institution_ids': [], 'apc_list': {'value': 2290, 'currency': 'EUR', 'value_usd': 2890, 'provenance': 'doaj'}, 'apc_paid': {'value': 2290, 'currency': 'EUR', 'value_usd': 2890, 'provenance': 'doaj'}, 'fwci': 1.129, 'has_fulltext': True, 'fulltext_origin': 'pdf', 'cited_by_count': 9, 'citation_normalized_percentile': {'value': 0.740337, 'is_in_top_1_percent': False, 'is_in_top_10_percent': False}, 'cited_by_percentile_year': {'min': 85, 'max': 86}, 'biblio': {'volume': '93', 'issue': '2', 'first_page': '347', 'last_page': '362'}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T12162', 'display_name': 'Cellular Automata and Applications', 'score': 1.0, 'subfield': {'id': 'https://openalex.org/subfields/1703', 'display_name': 'Computational Theory and Mathematics'}, 'field': {'id': 'https://openalex.org/fields/17', 'display_name': 'Computer Science'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, 'topics': [{'id': 'https://openalex.org/T12162', 'display_name': 'Cellular Automata and Applications', 'score': 1.0, 'subfield': {'id': 'https://openalex.org/subfields/1703', 'display_name': 'Computational Theory and Mathematics'}, 'field': {'id': 'https://openalex.org/fields/17', 'display_name': 'Computer Science'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T11567', 'display_name': 'semigroups and automata theory', 'score': 0.9846, 'subfield': {'id': 'https://openalex.org/subfields/1703', 'display_name': 'Computational Theory and Mathematics'}, 'field': {'id': 'https://openalex.org/fields/17', 'display_name': 'Computer Science'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T12029', 'display_name': 'DNA and Biological Computing', 'score': 0.991, 'subfield': {'id': 'https://openalex.org/subfields/1312', 'display_name': 'Molecular Biology'}, 'field': {'id': 'https://openalex.org/fields/13', 'display_name': 'Biochemistry, Genetics and Molecular Biology'}, 'domain': {'id': 'https://openalex.org/domains/1', 'display_name': 'Life Sciences'}}], 'keywords': [{'id': 'https://openalex.org/keywords/cardinality', 'display_name': 'Cardinality (data modeling)', 'score': 0.4657325}, {'id': 'https://openalex.org/keywords/rank', 'display_name': 'Rank (graph theory)', 'score': 0.44721133}], 'concepts': [{'id': 'https://openalex.org/C35527583', 'wikidata': 'https://www.wikidata.org/wiki/Q189156', 'display_name': 'Cellular automaton', 'level': 2, 'score': 0.7518599}, {'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.72883046}, {'id': 'https://openalex.org/C207405024', 'wikidata': 'https://www.wikidata.org/wiki/Q207348', 'display_name': 'Semigroup', 'level': 2, 'score': 0.5179295}, {'id': 'https://openalex.org/C114614502', 'wikidata': 'https://www.wikidata.org/wiki/Q76592', 'display_name': 'Combinatorics', 'level': 1, 'score': 0.49389082}, {'id': 'https://openalex.org/C118615104', 'wikidata': 'https://www.wikidata.org/wiki/Q121416', 'display_name': 'Discrete mathematics', 'level': 1, 'score': 0.48374748}, {'id': 'https://openalex.org/C87117476', 'wikidata': 'https://www.wikidata.org/wiki/Q362383', 'display_name': 'Cardinality (data modeling)', 'level': 2, 'score': 0.4657325}, {'id': 'https://openalex.org/C2781311116', 'wikidata': 'https://www.wikidata.org/wiki/Q83306', 'display_name': 'Group (periodic table)', 'level': 2, 'score': 0.4538998}, {'id': 'https://openalex.org/C184992742', 'wikidata': 'https://www.wikidata.org/wiki/Q7243229', 'display_name': 'Prime (order theory)', 'level': 2, 'score': 0.45349252}, {'id': 'https://openalex.org/C164226766', 'wikidata': 'https://www.wikidata.org/wiki/Q7293202', 'display_name': 'Rank (graph theory)', 'level': 2, 'score': 0.44721133}, {'id': 'https://openalex.org/C2777404646', 'wikidata': 'https://www.wikidata.org/wiki/Q1057968', 'display_name': 'Finite group', 'level': 3, 'score': 0.41593316}, {'id': 'https://openalex.org/C11413529', 'wikidata': 'https://www.wikidata.org/wiki/Q8366', 'display_name': 'Algorithm', 'level': 1, 'score': 0.13126877}, {'id': 'https://openalex.org/C41008148', 'wikidata': 'https://www.wikidata.org/wiki/Q21198', 'display_name': 'Computer science', 'level': 0, 'score': 0.1103138}, {'id': 'https://openalex.org/C185592680', 'wikidata': 'https://www.wikidata.org/wiki/Q2329', 'display_name': 'Chemistry', 'level': 0, 'score': 0.0}, {'id': 'https://openalex.org/C178790620', 'wikidata': 'https://www.wikidata.org/wiki/Q11351', 'display_name': 'Organic chemistry', 'level': 1, 'score': 0.0}, {'id': 'https://openalex.org/C124101348', 'wikidata': 'https://www.wikidata.org/wiki/Q172491', 'display_name': 'Data mining', 'level': 1, 'score': 0.0}], 'mesh': [], 'locations_count': 5, 'locations': [{'is_oa': True, 'landing_page_url': 'https://doi.org/10.1007/s00233-016-9783-z', 'pdf_url': 'https://link.springer.com/content/pdf/10.1007%2Fs00233-016-9783-z.pdf', 'source': {'id': 'https://openalex.org/S75897683', 'display_name': 'Semigroup Forum', 'issn_l': '0037-1912', 'issn': ['0037-1912', '1432-2137'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310319900', 'host_organization_name': 'Springer Science+Business Media', 'host_organization_lineage': ['https://openalex.org/P4310319965', 'https://openalex.org/P4310319900'], 'host_organization_lineage_names': ['Springer Nature', 'Springer Science+Business Media'], 'type': 'journal'}, 'license': 'cc-by', 'license_id': 'https://openalex.org/licenses/cc-by', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, {'is_oa': True, 'landing_page_url': 'https://arxiv.org/abs/1510.00197', 'pdf_url': 'https://arxiv.org/pdf/1510.00197', 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I205783295', 'host_organization_name': 'Cornell University', 'host_organization_lineage': ['https://openalex.org/I205783295'], 'host_organization_lineage_names': ['Cornell University'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://durham-repository.worktribe.com/file/1416579/1/Published%20Journal%20Article%20%28Advance%20online%20version%29', 'pdf_url': 'https://durham-repository.worktribe.com/file/1416579/1/Published%20Journal%20Article%20%28Advance%20online%20version%29', 'source': None, 'license': 'cc-by', 'license_id': 'https://openalex.org/licenses/cc-by', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, {'is_oa': True, 'landing_page_url': 'https://dro.dur.ac.uk/17994/3/17994P.pdf', 'pdf_url': 'https://durham-repository.worktribe.com/preview/1416591/17994.pdf', 'source': {'id': 'https://openalex.org/S4306400188', 'display_name': 'Durham Research Online (Durham University)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I190082696', 'host_organization_name': 'Durham University', 'host_organization_lineage': ['https://openalex.org/I190082696'], 'host_organization_lineage_names': ['Durham University'], 'type': 'repository'}, 'license': 'cc-by', 'license_id': 'https://openalex.org/licenses/cc-by', 'version': 'acceptedVersion', 'is_accepted': True, 'is_published': False}, {'is_oa': False, 'landing_page_url': 'https://api.datacite.org/dois/10.48550/arxiv.1510.00197', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4393179698', 'display_name': 'DataCite API', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I4210145204', 'host_organization_name': 'DataCite', 'host_organization_lineage': ['https://openalex.org/I4210145204'], 'host_organization_lineage_names': ['DataCite'], 'type': 'metadata'}, 'license': None, 'license_id': None, 'version': None}], 'best_oa_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1007/s00233-016-9783-z', 'pdf_url': 'https://link.springer.com/content/pdf/10.1007%2Fs00233-016-9783-z.pdf', 'source': {'id': 'https://openalex.org/S75897683', 'display_name': 'Semigroup Forum', 'issn_l': '0037-1912', 'issn': ['0037-1912', '1432-2137'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310319900', 'host_organization_name': 'Springer Science+Business Media', 'host_organization_lineage': ['https://openalex.org/P4310319965', 'https://openalex.org/P4310319900'], 'host_organization_lineage_names': ['Springer Nature', 'Springer Science+Business Media'], 'type': 'journal'}, 'license': 'cc-by', 'license_id': 'https://openalex.org/licenses/cc-by', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'sustainable_development_goals': [], 'grants': [], 'datasets': [], 'versions': ['https://openalex.org/W2234074323', 'https://openalex.org/W3104722361'], 'referenced_works_count': 24, 'referenced_works': ['https://openalex.org/W1493654001', 'https://openalex.org/W1584858174', 'https://openalex.org/W1711039925', 'https://openalex.org/W1977841303', 'https://openalex.org/W2000331453', 'https://openalex.org/W2005465537', 'https://openalex.org/W2013481822', 'https://openalex.org/W2014389955', 'https://openalex.org/W2019085230', 'https://openalex.org/W2022060148', 'https://openalex.org/W2066788715', 'https://openalex.org/W2073359254', 'https://openalex.org/W2079876213', 'https://openalex.org/W2101387847', 'https://openalex.org/W2108704847', 'https://openalex.org/W2313402717', 'https://openalex.org/W2324812597', 'https://openalex.org/W2488025056', 'https://openalex.org/W2798943694', 'https://openalex.org/W2963559008', 'https://openalex.org/W3104648411', 'https://openalex.org/W4234295943', 'https://openalex.org/W4292512159', 'https://openalex.org/W792370658'], 'related_works': ['https://openalex.org/W3091867061', 'https://openalex.org/W2360313292', 'https://openalex.org/W2359527201', 'https://openalex.org/W2353714615', 'https://openalex.org/W2164619678', 'https://openalex.org/W2113544496', 'https://openalex.org/W2067777412', 'https://openalex.org/W2058438338', 'https://openalex.org/W2002177687', 'https://openalex.org/W1669105318'], 'abstract_inverted_index': {'Since': [0], 'first': [1], 'introduced': [2], 'by': [3, 58, 112, 133], 'John': [4], 'von': [5], 'Neumann,': [6], 'the': [7, 36, 47, 54, 75, 86, 103, 125, 135, 138, 146, 156, 197], 'notion': [8], 'of': [9, 35, 38, 41, 49, 53, 77, 83, 85, 105, 130, 140, 145, 151], 'cellular': [10, 30, 106, 131, 153], 'automaton': [11, 31], 'has': [12, 108], 'grown': [13], 'into': [14], 'a': [15, 29, 33, 42, 59, 67, 78, 141, 161], 'key': [16], 'concept': [17], 'in': [18, 81], 'computer': [19], 'science,': [20], 'physics': [21], 'and': [22, 98, 118, 160, 185, 189, 193], 'theoretical': [23], 'biology.': [24], 'In': [25, 71, 120, 165], 'its': [26, 113], 'classical': [27], 'setting,': [28], 'is': [32, 56, 95, 100, 173], 'transformation': [34], 'set': [37, 163], 'all': [39, 152], 'configurations': [40], 'regular': [43], 'grid': [44, 55], 'such': [45], 'that': [46, 63], 'image': [48], 'any': [50, 96, 101, 181], 'particular': [51], 'cell': [52], 'determined': [57], 'fixed': [60, 68], 'local': [61], 'function': [62], 'only': [64], 'depends': [65], 'on': [66], 'finite': [69, 126, 162], 'neighbourhood.': [70], 'recent': [72], 'years,': [73], 'with': [74, 115], 'introduction': [76], 'generalised': [79], 'definition': [80], 'terms': [82], 'transformations': [84], 'form': [87], '$\\tau': [88], ':': [89], 'A^G': [90], '\\to': [91], 'A^G$': [92], '(where': [93], '$G$': [94], 'group': [97, 116, 158], '$A$': [99], 'set),': [102], 'theory': [104, 117], 'automata': [107, 132, 154], 'been': [109], 'greatly': [110], 'enriched': [111], 'connections': [114], 'topology.': [119], 'this': [121, 169], 'paper,': [122], 'we': [123, 167, 190], 'begin': [124], 'semigroup': [127, 147], 'theoretic': [128], 'study': [129], 'investigating': [134], 'rank': [136, 170], '(i.e.': [137], 'cardinality': [139], 'smallest': [142], 'generating': [143], 'set)': [144], '$\\text{CA}(\\mathbb{Z}_n;': [148], 'A)$': [149], 'consisting': [150], 'over': [155], 'cyclic': [157], '$\\mathbb{Z}_n$': [159], '$A$.': [164], 'particular,': [166], 'determine': [168], 'when': [171], '$n$': [172], 'equal': [174], 'to': [175], '$p$,': [176], '$2^k$': [177], 'or': [178], '$2^kp$,': [179], 'for': [180, 196], 'odd': [182], 'prime': [183], '$p$': [184], '$k': [186], '\\geq': [187], '1$,': [188], 'give': [191], 'upper': [192], 'lower': [194], 'bounds': [195], 'general': [198], 'case.': [199]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W3104722361', 'counts_by_year': [{'year': 2021, 'cited_by_count': 1}, {'year': 2020, 'cited_by_count': 3}, {'year': 2019, 'cited_by_count': 2}, {'year': 2017, 'cited_by_count': 2}, {'year': 2016, 'cited_by_count': 1}], 'updated_date': '2024-12-11T19:39:29.536823', 'created_date': '2020-11-23'}