{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:06:59Z","timestamp":1760234819539,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,6,25]],"date-time":"2021-06-25T00:00:00Z","timestamp":1624579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61741307; 61903037"],"award-info":[{"award-number":["61741307; 61903037"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2020M670164"],"award-info":[{"award-number":["2020M670164"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.<\/jats:p>","DOI":"10.3390\/sym13071137","type":"journal-article","created":{"date-parts":[[2021,6,25]],"date-time":"2021-06-25T11:07:40Z","timestamp":1624619260000},"page":"1137","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Ordered Structures of Polynomials over Max-Plus Algebra"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6300-8972","authenticated-orcid":false,"given":"Cailu","family":"Wang","sequence":"first","affiliation":[{"name":"School of Automation, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanqing","family":"Xia","sequence":"additional","affiliation":[{"name":"School of Automation, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4760-0591","authenticated-orcid":false,"given":"Yuegang","family":"Tao","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Hebei University of Technology, Tianjin 300130, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cuninghame-Green, R.A. (1979). Minimax Algebra, Springer.","DOI":"10.1007\/978-3-642-48708-8"},{"key":"ref_2","unstructured":"Baccelli, F., Cohen, G., Olsder, G.J., and Quadrat, J.P. (1992). Synchronization and Linearity: An Algebra for Discrete Event System, John Wiley and Sons."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Heidergott, B., Olsder, G.J., and van der Woude, J. (2006). Max-Plus at Work: Modeling and Analysis of Synchronized Systems, Princeton University Press.","DOI":"10.1515\/9781400865239"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Butkovi\u010d, P. (2010). Max-Linear Systems: Theory and Algorithms, Springer.","DOI":"10.1007\/978-1-84996-299-5"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1109\/TAC.1985.1103925","article-title":"A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing","volume":"30","author":"Cohen","year":"1985","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.dam.2013.08.007","article-title":"On the integer max-linear programming problem","volume":"162","author":"MacCaig","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1109\/TAC.2016.2604562","article-title":"Observer-based controllers for max-plus linear systems","volume":"62","author":"Hardouin","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.automatica.2017.11.030","article-title":"Conditional reachability of uncertain max plus linear systems","volume":"94","author":"Candido","year":"2018","journal-title":"Automatica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.automatica.2019.05.026","article-title":"On max-plus linear dynamical system theory: The observation problem","volume":"107","author":"Goncalves","year":"2019","journal-title":"Automatica"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1109\/TAC.2018.2849570","article-title":"Model predictive control for stochast max-plus linear systems with chance constraints","volume":"64","author":"Xu","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Tao, Y., and Wang, C. (2020). Global optimization for max-plus linear systems and applications in distributed systems. Automatica, 119.","DOI":"10.1016\/j.automatica.2020.109104"},{"key":"ref_12","unstructured":"Gaubert, S. (1992). Th\u00e9orie des Syst\u00e8mes Lin\u00e9aires dans les Dioides. [Ph.D. Thesis, L\u2019Ecole Nationale Sup\u00e9rieure des Mines de Paris]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0024-3795(93)90492-7","article-title":"The power algorithm in max algebra","volume":"182","author":"Braker","year":"1993","journal-title":"Linear Algebra Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10626-012-0137-z","article-title":"The level set method for the two-sided max-plus eigenproblem","volume":"23","author":"Gaubert","year":"2013","journal-title":"Discret. Event Dyn. Syst. Theory Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1002","DOI":"10.1137\/15M1041031","article-title":"On special cases of the generalized max-plus eigenproblem","volume":"37","author":"Jones","year":"2016","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Umer, M., Hayat, U., and Abbas, F. (2019). An efficient algorithm for nontrivial eigenvectors in max-plus algebra. Symmetry, 11.","DOI":"10.3390\/sym11060738"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Umer, M., Hayat, U., Abbas, F., Agarwal, A., and Kitanov, P. (2020). An efficient algorithm for eigenvalue problem of latin squares in a bipartite min-max-plus system. Symmetry, 12.","DOI":"10.3390\/sym12020311"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0024-3795(88)90145-0","article-title":"Cramer and Cayley-Hamilton in the max algebra","volume":"101","author":"Olsder","year":"1988","journal-title":"Linear Algebra Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/S0024-3795(96)00407-7","article-title":"A note on the characteristic equation in the max-plus algebra","volume":"261","year":"1997","journal-title":"Linear Algebra Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1137\/S0895479896304782","article-title":"The QR decomposition and the singular value decomposition in the symmetrized max-plus algebra","volume":"19","year":"1998","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.laa.2015.10.031","article-title":"Interval max-plus matrix equations","volume":"492","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.cam.2016.11.027","article-title":"Closed form of L-localized solution set of max-plus interval linear system and its application on optimization problem","volume":"317","author":"Thipwiwatpotjana","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.laa.2017.10.001","article-title":"Interval strong solutions of interval systems of max-plus linear equations","volume":"537","author":"Wang","year":"2018","journal-title":"Linear Algebra Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/0166-218X(80)90025-6","article-title":"An algebra for piecewise-linear minimax problems","volume":"2","author":"Meijer","year":"1980","journal-title":"Discret. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/0165-0114(95)00012-A","article-title":"Maxpolynomial equations","volume":"75","year":"1995","journal-title":"Fuzzy Sets Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/BF01797235","article-title":"A method to find all solutions of a system of multivariate polynomial equalities and inequalities in the max algebra","volume":"6","year":"1996","journal-title":"Discret. Event Dyn. Syst. Theory Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0166-218X(03)00223-3","article-title":"Finding all essential terms of a characteristic maxpolynomial","volume":"130","author":"Burkard","year":"2003","journal-title":"Discret. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s10107-003-0411-9","article-title":"Max algebra and the linear assignment problem","volume":"98","author":"Burkard","year":"2003","journal-title":"Math. Program. Ser. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"489","DOI":"10.4171\/jems\/309","article-title":"Idempotent semigroups and tropical algebraic sets","volume":"14","author":"Izhakian","year":"2012","journal-title":"J. Eur. Math. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3262","DOI":"10.1016\/j.jpaa.2018.11.001","article-title":"Summand absorbing submodules of a module over a semiring","volume":"223","author":"Izhakian","year":"2019","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1016\/j.laa.2019.05.020","article-title":"Polynomial convolutions in max-plus algebra","volume":"578","author":"Rosenmann","year":"2019","journal-title":"Linear Algebra Appl."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wang, C., and Tao, Y. (2021). Matrix representation of formal polynomials over max-plus algebra. J. Algebra Appl.","DOI":"10.1142\/S0219498821502169"},{"key":"ref_33","unstructured":"Fuchs, L. (1963). Partially Ordered Algebraic Systems, Pergamon Press."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Golan, J.S. (2003). Semirings and Affine Equations over Them: Theory and Applications, Kluwer Academic.","DOI":"10.1007\/978-94-017-0383-3"},{"key":"ref_35","unstructured":"Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures, Springer."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5563","DOI":"10.1081\/AGB-120023974","article-title":"Quotient ordered homomorphisms of ordered semigroups","volume":"13","author":"Cao","year":"2003","journal-title":"Commun. Algebra"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1016\/j.laa.2016.06.001","article-title":"Minimal orderings and quadratic forms on a free module over a supertropical semiring","volume":"507","author":"Izhakian","year":"2016","journal-title":"Linear Algebra Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1137\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:24:14Z","timestamp":1760163854000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1137"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,25]]},"references-count":37,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071137"],"URL":"https:\/\/doi.org\/10.3390\/sym13071137","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,6,25]]}}}