{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T07:43:06Z","timestamp":1772437386612,"version":"3.50.1"},"reference-count":16,"publisher":"Canadian Mathematical Society","issue":"1","license":[{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"unspecified","delay-in-days":11980,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Can. j. math."],"published-print":{"date-parts":[[1986,2,1]]},"abstract":"The most complete work on the structure of the lattice of varieties of commutative semigroups available to this date is [12]<\/jats:bold>. Nevertheless, it fails to give the structure of this lattice. In the positive direction, it shows in particular that the order structure of is determined by the order structure of well-known lattices of integers together with the sublattice of varieties of commutative nil semigroups.<\/jats:p>In the present work, we study from the point of view of order. Perkins [13]<\/jats:bold> has shown that has no infinite descending chains and is countable. The underlying questions we consider here arose from the results of Almeida and Reilly [1<\/jats:bold>] in connection with generalized varieties. There, it is observed that the best-known part of consisting of the\nvarieties all of whose elements are abelian groups is in a sense very wide: it contains infinite subsets of mutually incomparable elements and allows the construction of uncountably many generalized varieties and infinite descending chains of generalized varieties.<\/jats:p>","DOI":"10.4153\/cjm-1986-002-x","type":"journal-article","created":{"date-parts":[[2010,12,7]],"date-time":"2010-12-07T18:24:44Z","timestamp":1291746284000},"page":"19-47","source":"Crossref","is-referenced-by-count":13,"title":["Some Order Properties of the Lattice of Varieties of Commutative Semigroups"],"prefix":"10.4153","volume":"38","author":[{"given":"Jorge","family":"Almeida","sequence":"first","affiliation":[]}],"member":"2643","published-online":{"date-parts":[[2018,11,20]]},"reference":[{"key":"S0008414X00006520_R0045","first-page":"591","volume-title":"Commutative semigroup laws","volume":"22","author":"Schwabauer","year":"1969"},{"key":"S0008414X00006520_R0042","first-page":"298","volume-title":"Bases for equational theories of semi-groups","volume":"11","author":"Perkins","year":"1969"},{"key":"S0008414X00006520_R0038","first-page":"297","volume-title":"The theory of well-quasi-ordering: a frequently discovered concept","volume":"13","author":"Kruskal","year":"1972"},{"key":"S0008414X00006520_R0037","first-page":"285","volume-title":"A sketch of the lattice of commutative nilpotent semigroups varieties","volume":"24","author":"Korjakov","year":"1982"},{"key":"S0008414X00006520_R0034","volume-title":"Automata, languages and machines","volume":"B","author":"Eilenberg","year":"1976"},{"key":"S0008414X00006520_R0043","volume-title":"Infinite iteration of matrix semigroups, Part II: Structure theorem for arbitrary semigroups up to aperiodic morphism","author":"Rhodes","year":"1983"},{"key":"S0008414X00006520_R0036","first-page":"326","volume-title":"Ordering by divisibility in abstract algebras","volume":"2","author":"Higman","year":"1952"},{"key":"S0008414X00006520_R0039","first-page":"31","volume-title":"Better-quasi-orderings and a class of trees in Studies in foundations and combinatorics","volume":"1","author":"Laver","year":"1978"},{"key":"S0008414X00006520_R0032","first-page":"189","volume-title":"On the cardinality of the set of initial intervals of a partially ordered set","volume":"10","author":"Bonnet","year":"1973"},{"key":"S0008414X00006520_R0030","first-page":"77","volume-title":"Generalized varieties of commutative and nilpotent semigroups","volume":"30","author":"Almeida","year":"1984"},{"key":"S0008414X00006520_R0041","first-page":"875","volume-title":"The lattice of equational classes of commutative semigroups","volume":"23","author":"Nelson","year":"1971"},{"key":"S0008414X00006520_R0040","first-page":"697","volume-title":"On well-quasi-ordering infinite trees","volume":"61","author":"Nash-Williams","year":"1965"},{"key":"S0008414X00006520_R0035","first-page":"413","volume-title":"On pseudovarieties","volume":"19","author":"Eilenberg","year":"1976"},{"key":"S0008414X00006520_R0044","first-page":"503","volume-title":"A note on commutative semigroups","volume":"20","author":"Schwabauer","year":"1969"},{"key":"S0008414X00006520_R0033","first-page":"37","volume-title":"Embeddings \u03a0m in the lattice equational classes of commutative semigroups","volume":"30","author":"Burris","year":"1971"},{"key":"S0008414X00006520_R0031","unstructured":"Ash C. J. , Pseudovarieties, generalized varieties and similarly defined classes, to appear in J. Algebra."}],"container-title":["Canadian Journal of Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0008414X00006520","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T15:15:49Z","timestamp":1558538149000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0008414X00006520\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,2,1]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1986,2,1]]}},"alternative-id":["S0008414X00006520"],"URL":"https:\/\/doi.org\/10.4153\/cjm-1986-002-x","relation":{},"ISSN":["0008-414X","1496-4279"],"issn-type":[{"value":"0008-414X","type":"print"},{"value":"1496-4279","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,2,1]]}}}