{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T16:12:35Z","timestamp":1760199155627,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,6]],"date-time":"2018-11-06T00:00:00Z","timestamp":1541462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Princess Nourah bint Abdulrahman University.","award":["261-s-39"],"award-info":[{"award-number":["261-s-39"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper concerns three relationship between the recently proposed cubic sets and finite state machines. The notions of cubic finite state machine (cubic FSM), a subsystem of cubic FSM and cartesian composition (direct product, P-(R-) union, and P-(R-) intersection) of two subsystems of cubic FSMs are introduced. We study the cartesian composition, direct product and union of two subsystems of cubic FSMs is a subsystem of a cubic FSM. We provide many examples on each case. We consider conditions for subsystem of cubic FSM to be both an internal cubic subsystem of cubic FSM and an external cubic subsystem of cubic FSM.<\/jats:p>","DOI":"10.3390\/sym10110598","type":"journal-article","created":{"date-parts":[[2018,11,7]],"date-time":"2018-11-07T03:45:22Z","timestamp":1541562322000},"page":"598","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Applications of Cubic Structures to Subsystems of Finite State Machines"],"prefix":"10.3390","volume":"10","author":[{"given":"Nabilah","family":"Abughazalah","sequence":"first","affiliation":[{"name":"Mathematical Sciences Department, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Naveed","family":"Yaqoob","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi 11932, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning-I","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"24","DOI":"10.20852\/ntmsci.2017.211","article-title":"Cubic finite state machine and cubic transformation semigroups","volume":"5","author":"Abdullah","year":"2017","journal-title":"New Trends Math. Sci."},{"key":"ref_3","first-page":"239","article-title":"The Cartesian composition of automata","volume":"11","year":"1978","journal-title":"Math. Syst. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.3233\/IFS-151642","article-title":"Soft finite state machine","volume":"29","author":"Hussain","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF02936044","article-title":"Intuitionistic fuzzy finite state machines","volume":"17","author":"Jun","year":"2005","journal-title":"J. Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF02831941","article-title":"Intuitionistic fuzzy finite switchboard state machines","volume":"20","author":"Jun","year":"2006","journal-title":"J. Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4977","DOI":"10.1016\/j.ins.2007.06.008","article-title":"Quotient structures of intuitionistic fuzzy finite state machines","volume":"177","author":"Jun","year":"2007","journal-title":"Inf. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4284","DOI":"10.1016\/j.ins.2009.08.017","article-title":"Intuitionistic fuzzy transformation semigroups","volume":"179","author":"Jun","year":"2009","journal-title":"Inf. Sci."},{"key":"ref_9","first-page":"83","article-title":"Cubic sets","volume":"4","author":"Jun","year":"2012","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3334","DOI":"10.1016\/j.camwa.2011.08.042","article-title":"Cubic structures applied to ideals of BCI-algebras","volume":"62","author":"Jun","year":"2011","journal-title":"Comp. Math. Appl."},{"key":"ref_11","first-page":"749","article-title":"The properties on fuzzy submachines of a fuzzy finite state machine","volume":"13","author":"Hwang","year":"2003","journal-title":"J. Korean Inst. Intell. Syst."},{"key":"ref_12","first-page":"181","article-title":"Bipolar fuzzy finite state machines","volume":"34","author":"Jun","year":"2011","journal-title":"Bull. Malaysian Math. Sci. Soc."},{"key":"ref_13","first-page":"233","article-title":"N-structures applied to finite state machines","volume":"43","author":"Kavikumar","year":"2013","journal-title":"IAENG Int. J. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1361","DOI":"10.1007\/s13370-016-0416-1","article-title":"Interval neutrosophic finite switchboard state machine","volume":"27","author":"Mahmood","year":"2016","journal-title":"Afr. Mat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0165-0114(01)00059-8","article-title":"On covering of products of fuzzy finite state machines","volume":"125","author":"Kumbhojkar","year":"2002","journal-title":"Fuzzy Sets Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1108\/03684929510089394","article-title":"The Cartesian composition of fuzzy finite state machines","volume":"24","author":"Malik","year":"1995","journal-title":"Kybernetes"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0165-0114(94)90274-7","article-title":"On subsystems of a fuzzy finite state machine","volume":"68","author":"Malik","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","first-page":"87","article-title":"Semigroups of fuzzy finite state machines","volume":"2","author":"Malik","year":"1994","journal-title":"Adv. Fuzzy Theory Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0165-0114(96)00166-2","article-title":"Products of fuzzy finite state machines","volume":"92","author":"Malik","year":"1997","journal-title":"Fuzzy Sets Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.1016\/j.fss.2009.01.001","article-title":"Products of Mealy-type fuzzy finite state machines","volume":"160","author":"Liu","year":"2009","journal-title":"Fuzzy Sets Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/11\/598\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:28:11Z","timestamp":1760196491000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/11\/598"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,6]]},"references-count":20,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["sym10110598"],"URL":"https:\/\/doi.org\/10.3390\/sym10110598","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,11,6]]}}}