{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T09:12:39Z","timestamp":1779354759663,"version":"3.51.4"},"reference-count":26,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:00:00Z","timestamp":1777420800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["APVV-20-0069"],"award-info":[{"award-number":["APVV-20-0069"]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["VEGA No. 2\/0128\/24 SAV, A.D."],"award-info":[{"award-number":["VEGA No. 2\/0128\/24 SAV, A.D."]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,4,29]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We investigate nodes (elements that are comparable with all other elements) and their role in the structural analysis of pseudo-hoops. First, we study idempotent nodes and present a complete representation of pseudo-hoops with finitely many idempotent nodes, based on their decomposition into ordinal sums of pseudo-hoops. Next, we examine ordinal sum irreducible pseudo-hoops and prove that every Wajsberg pseudo-hoop with a non-trivial node is linearly ordered. Moreover, we show that the set of nodes $\\text{Nod}(\\mathbf E)$ of the ordinal sum of any family of Wajsberg pseudo-hoops forms a subalgebra. We then use the set of regular elements of a bounded representable pseudo-hoop $\\mathbf E$ and its relationship with $\\text{Nod}(\\mathbf E)$ to determine whether $\\mathbf E$ is ordinal sum reducible or irreducible. As a consequence, we establish that every finite basic pseudo-hoop can be expressed as an ordinal sum of a finite family ${\\mathbf E_{1},\\ldots ,\\mathbf E_{n}}$ for some $n\\in \\mathbb N$, where $\\mathbf E_{1},\\ldots ,\\mathbf E_{n-1}$ are linear Wajsberg pseudo-hoops and $\\mathbf E_{n}$ is ordinal sum irreducible.<\/jats:p>","DOI":"10.1093\/logcom\/exag022","type":"journal-article","created":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T11:34:54Z","timestamp":1777376094000},"source":"Crossref","is-referenced-by-count":0,"title":["Nodes of pseudo-hoops"],"prefix":"10.1093","volume":"36","author":[{"given":"Anatolij","family":"Dvure\u010denskij","sequence":"first","affiliation":[{"name":"Mathematical Institute, Slovak Academy of Sciences , \u0160tef\u00e1nikova 49, SK-814 73 Bratislava ,","place":["Slovakia"]},{"name":"Faculty of Sciences, Palack\u00fd University Olomouc , t\u0159. 17. listopadu 12, CZ-771 46 Olomouc ,","place":["Czech Republic"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Omid","family":"Zahiri","sequence":"additional","affiliation":[{"name":"Independent Scholar , Tehran ,","place":["Iran"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2026,5,21]]},"reference":[{"key":"2026052104555108400_ref1","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/S0022-4049(02)00329-8","article-title":"Varieties of BL-algebras I: general properties","volume":"181","author":"Aglian\u00f2","year":"2003","journal-title":"Journal of Pure and Applied Algebra"},{"key":"2026052104555108400_ref2","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s11225-007-9078-1","article-title":"Basic hoops: an algebraic study of continuous t-norms","volume":"87","author":"Aglian\u00f2","year":"2007","journal-title":"Studia Logica"},{"key":"2026052104555108400_ref3","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-2871-8","volume-title":"Lattice-Ordered Groups: An Introduction","author":"Anderson","year":"1988"},{"key":"2026052104555108400_ref4","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s000120050156","article-title":"On the structure of hoops","volume":"43","author":"Blok","year":"2000","journal-title":"Algebra Universalis"},{"key":"2026052104555108400_ref5","doi-asserted-by":"crossref","first-page":"257","DOI":"10.4064\/fm-64-3-257-287","article-title":"Komplement\u00e4re Halbgruppen. 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