{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:44:47Z","timestamp":1760233487435,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,16]],"date-time":"2021-01-16T00:00:00Z","timestamp":1610755200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["M 2967"],"award-info":[{"award-number":["M 2967"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A subset A of a semigroup S is called a chain (antichain) if ab\u2208{a,b} (ab\u2209{a,b}) for any (distinct) elements a,b\u2208A. A semigroup S is called periodic if for every element x\u2208S there exists n\u2208N such that xn is an idempotent. A semigroup S is called (anti)chain-finite if S contains no infinite (anti)chains. We prove that each antichain-finite semigroup S is periodic and for every idempotent e of S the set e\u221e={x\u2208S:\u2203n\u2208N(xn=e)} is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Furthermore, we present an example of an antichain-finite semilattice that is not a union of finitely many chains.<\/jats:p>","DOI":"10.3390\/axioms10010009","type":"journal-article","created":{"date-parts":[[2021,1,18]],"date-time":"2021-01-18T05:17:34Z","timestamp":1610947054000},"page":"9","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6828-1193","authenticated-orcid":false,"given":"Iryna","family":"Banakh","sequence":"first","affiliation":[{"name":"Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Naukova 3b, 79060 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6710-4611","authenticated-orcid":false,"given":"Taras","family":"Banakh","sequence":"additional","affiliation":[{"name":"Faculty of Mehcanics and Mathematics, Ivan Franko National University of Lviv, Universytetska 1, 79000 Lviv, Ukraine"},{"name":"Katedra Matematyki, Jan Kochanowski University in Kielce, Universytecka 7, 25-406 Kielce, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2266-2024","authenticated-orcid":false,"given":"Serhii","family":"Bardyla","sequence":"additional","affiliation":[{"name":"Kurt G\u00f6del Research Center, Institute of Mathematics, University of Vienna, 1090 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J., Mislove, M., and Scott, D. (2003). Continuous Lattices and Domains, Cambridge University Press. Encyclopedia of Mathematics and its Applications, 93.","DOI":"10.1017\/CBO9780511542725"},{"key":"ref_2","first-page":"423","article-title":"Infinite antichains and duality theories","volume":"14","author":"Lawson","year":"1988","journal-title":"Houston J. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"243","DOI":"10.2140\/pjm.1975.58.243","article-title":"Algebraic maximal semilattices","volume":"58","author":"Stepp","year":"1975","journal-title":"Pacific J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1007\/s00233-018-9921-x","article-title":"Characterizing Chain-Finite and Chain-Compact Topological Semilattices","volume":"98","author":"Banakh","year":"2019","journal-title":"Semigroup Forum"},{"key":"ref_5","first-page":"177","article-title":"Complete Topologized Posets and Semilattices","volume":"57","author":"Banakh","year":"2021","journal-title":"Topol. Proc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s11083-010-9140-x","article-title":"On chains in H-closed topological pospaces","volume":"27","author":"Gutik","year":"2010","journal-title":"Order"},{"key":"ref_7","first-page":"287","article-title":"On the relation between completeness and H-closedness of pospaces without infinite antichains","volume":"15","author":"Yokoyama","year":"2013","journal-title":"Algebra Discr. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Clifford, A., and Preston, G. (1961). The Algebraic Theory of Semigroups, American Mathematical Soc.","DOI":"10.1090\/surv\/007.1"},{"key":"ref_9","unstructured":"Petrich, M., and Reilly, N. (1999). Completely Regular Semigroups, John Wiley & Sons, Inc."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Graham, R., Rothschild, B., and Spencer, J. (1990). Ramsey Theory, John Wiley & Sons, Inc.","DOI":"10.1038\/scientificamerican0790-112"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:11:52Z","timestamp":1760159512000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,16]]},"references-count":10,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["axioms10010009"],"URL":"https:\/\/doi.org\/10.3390\/axioms10010009","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2021,1,16]]}}}