{"id":360542,"date":"2017-02-01T15:56:22","date_gmt":"2017-02-01T23:56:22","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=360542"},"modified":"2018-10-16T20:04:32","modified_gmt":"2018-10-17T03:04:32","slug":"word-problem-cancellation-semigroups-zero","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/word-problem-cancellation-semigroups-zero\/","title":{"rendered":"The Word Problem for Cancellation Semigroups with Zero"},"content":{"rendered":"<p>In 1947, Post showed the word problem for semigroups to be undecidable. In 1950, Turing strengthened this result to cancellation semigroups, i.e. semigroups satisfying the cancellation property<br \/>\n(1)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 if xy = xz or yx = zx then y = z.<br \/>\nNo semigroups with zero satisfies (1). The cancellation property for semigroups with zero and identity is<br \/>\n(2)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 if xy = xz \\neq 0 or yx = zx \\neq 0 then y = z.<br \/>\nThe cancellation property for semigroups with zero bur without identity is the conjunction of (2) and<br \/>\n(3)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 if xy = x or yx = x then x = 0.<br \/>\nWhether or not a semigroup with zero has an identity, we refer to it as a cancellation semigroup with zero if it satisfies the appropriate cancellation property. It is shown in <a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"http:\/\/web.eecs.umich.edu\/~gurevich\/annotated.htm#8\">8<span class=\"sr-only\"> (opens in new tab)<\/span><\/a>, that the word problem for finite semigroups is undecidable. Here we show that the word problem is undecidable for finite cancellation semigroups with zero; this holds for semigroups with identity and also for semigroups wihtout identity. (In fact, we prove a stronger effective inseparabilit result.) This provides the necessary mathematical foundation for <a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"http:\/\/web.eecs.umich.edu\/~gurevich\/annotated.htm#41\">41<span class=\"sr-only\"> (opens in new tab)<\/span><\/a>).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In 1947, Post showed the word problem for semigroups to be undecidable. In 1950, Turing strengthened this result to cancellation semigroups, i.e. semigroups satisfying the cancellation property (1)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 if xy = xz or yx = zx then y = z. No semigroups with zero satisfies (1). The cancellation property for semigroups with zero and identity [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Journal of Symbolic Logic","msr_number":"","msr_organization":"","msr_pages_string":"184-191","msr_page_range_start":"184","msr_page_range_end":"191","msr_series":"","msr_volume":"49","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"1984-07-07","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/53.pdf","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13560],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-360542","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-programming-languages-software-engineering","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"1984-07-07","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"184-191","msr_chapter":"","msr_isbn":"","msr_journal":"Journal of Symbolic Logic","msr_volume":"49","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/53.pdf","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/53.pdf","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/53.pdf"}],"msr-author-ordering":[{"type":"user_nicename","value":"gurevich","user_id":31929,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=gurevich"},{"type":"text","value":"H. 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