{"title":"On transitive Cayley graphs of homogeneous inverse semigroups","authors":"E. Ilić-Georgijević","doi":"10.1007/s10474-023-01375-x","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>S</i> be a pseudo-unitary homogeneous (graded) inverse semigroup\nwith zero 0, that is, an inverse semigroup with zero, and with a family <span>\\(\\{S_\\delta\\}_{\\delta\\in\\Delta}\\)</span> of nonzero subsets of <i>S</i>, called components of <i>S</i>,\nindexed by a partial groupoid <span>\\(\\Delta\\)</span>, that is, by a set with a partial binary operation, such that\n<span>\\(S=\\bigcup_{\\delta\\in\\Delta}S_\\delta\\)</span>, \nand: i) <span>\\(S_\\xi\\cap S_\\eta\\subseteq\\{0\\}\\)</span> for all distinct <span>\\(\\xi,\\eta\\in\\Delta;\\)</span>\nii) <span>\\(S_\\xi S_\\eta\\subseteq S_{\\xi\\eta}\\)</span> whenever <span>\\(\\xi\\eta\\)</span> is defined;\niii) <span>\\(S_\\xi S_\\eta\\nsubseteq\\{0\\}\\)</span> if and only if the product <span>\\(\\xi\\eta\\)</span> is defined;\niv) for every idempotent element <span>\\(\\epsilon\\in\\Delta\\)</span>, the subsemigroup <span>\\(S_\\epsilon\\)</span> is with identity <span>\\(1_\\epsilon;\\)</span>\nv) for every <span>\\(x\\in S\\)</span> there exist idempotent elements <span>\\(\\xi, \\eta\\in\\Delta\\)</span> such that <span>\\(1_\\xi x=x=x1_\\eta;\\)</span>\nvi) <span>\\(1_\\xi1_\\eta=1_{\\xi\\eta}\\)</span> whenever <span>\\(\\xi\\eta\\in\\Delta\\)</span> is an idempotent element, where <span>\\(\\xi\\)</span>, <span>\\(\\eta\\)</span> are idempotent elements of <span>\\(\\Delta\\)</span>.\nLet <i>A</i> be a subset of the union of the subsemigroup components of <i>S</i>, which does not contain 0. By <span>\\(\\operatorname{Cay}(S^*,A)\\)</span> we denote a graph obtained \nfrom the Cayley graph <span>\\(\\operatorname{Cay}(S,A)\\)</span> by removing 0 and its incident\nedges. We characterize vertex-transitivity of <span>\\(\\operatorname{Cay}(S^*,A)\\)</span> and relate it \nto the vertex-transitivity of its subgraph whose vertex set is <span>\\(S_\\mu\\setminus\\{0\\}\\)</span>, where <span>\\(\\mu\\)</span> is the maximum element of the set of all idempotent elements of <span>\\(\\Delta\\)</span>,\nwith respect to the natural order.\n</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-023-01375-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let S be a pseudo-unitary homogeneous (graded) inverse semigroup
with zero 0, that is, an inverse semigroup with zero, and with a family \(\{S_\delta\}_{\delta\in\Delta}\) of nonzero subsets of S, called components of S,
indexed by a partial groupoid \(\Delta\), that is, by a set with a partial binary operation, such that
\(S=\bigcup_{\delta\in\Delta}S_\delta\),
and: i) \(S_\xi\cap S_\eta\subseteq\{0\}\) for all distinct \(\xi,\eta\in\Delta;\)
ii) \(S_\xi S_\eta\subseteq S_{\xi\eta}\) whenever \(\xi\eta\) is defined;
iii) \(S_\xi S_\eta\nsubseteq\{0\}\) if and only if the product \(\xi\eta\) is defined;
iv) for every idempotent element \(\epsilon\in\Delta\), the subsemigroup \(S_\epsilon\) is with identity \(1_\epsilon;\)
v) for every \(x\in S\) there exist idempotent elements \(\xi, \eta\in\Delta\) such that \(1_\xi x=x=x1_\eta;\)
vi) \(1_\xi1_\eta=1_{\xi\eta}\) whenever \(\xi\eta\in\Delta\) is an idempotent element, where \(\xi\), \(\eta\) are idempotent elements of \(\Delta\).
Let A be a subset of the union of the subsemigroup components of S, which does not contain 0. By \(\operatorname{Cay}(S^*,A)\) we denote a graph obtained
from the Cayley graph \(\operatorname{Cay}(S,A)\) by removing 0 and its incident
edges. We characterize vertex-transitivity of \(\operatorname{Cay}(S^*,A)\) and relate it
to the vertex-transitivity of its subgraph whose vertex set is \(S_\mu\setminus\{0\}\), where \(\mu\) is the maximum element of the set of all idempotent elements of \(\Delta\),
with respect to the natural order.
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.