### From pre-trusses to skew braces

Tomasz Brzezinski , Stefano Mereta, Bernard Rybolowicz

Publicacions Matemàtiques, Volume: 66, Issue: 2, Pages: 683 - 714

Swansea University Authors:

• PDF | Accepted Manuscript

Released with publisher's permission.

DOI (Published version): 10.5565/publmat6622206

Abstract

An algebraic system consisting of a set together with an associative binary and a ternary heap operations is studied. Such a system is termed a pre-truss and if a binary operation distributes over the heap operation on one side one speaks about a near-truss. If the binary operation in a near-truss i...

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Published in: Publicacions Matemàtiques 0214-1493 2014-4369 Universitat Autonoma de Barcelona 2022 https://cronfa.swan.ac.uk/Record/cronfa56171 No Tags, Be the first to tag this record!
first_indexed 2021-02-03T18:33:11Z 2023-01-11T14:35:13Z cronfa56171 SURis 2022-09-28T13:06:30.2858622v2561712021-02-03From pre-trusses to skew braces30466d840b59627325596fbbb2c827540000-0001-6270-3439TomaszBrzezinskiTomasz Brzezinskitruefalse73f1af0073c82ade8fb20d1550e505d8StefanoMeretaStefano Meretatruefalsec821b8f926d58c3b7955b533a5c1472bBernardRybolowiczBernard Rybolowicztruefalse2021-02-03SMAAn algebraic system consisting of a set together with an associative binary and a ternary heap operations is studied. Such a system is termed a pre-truss and if a binary operation distributes over the heap operation on one side one speaks about a near-truss. If the binary operation in a near-truss is a group operation, then it can be specified or retracted to a skew brace, the notion introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534]. On the other hand if the binary operation in a near-truss has identity, then it gives rise to a skew-ring as introduced in [W. Rump, J. Algebra Appl. 18 (2019), 1950145]. Congruences in pre- and near-trusses are shown to arise from normal sub-heaps with an additional closure property of equivalence classes that involves both the ternary and binary operations. Such sub-heaps are called paragons. A necessary and sufficient criterion on paragonsunder which the quotient of a unital near-truss corresponds to a skew brace is derived. Regular elements in a pre-truss are defined as elements with left and right cancellation properties; following the ring-theoretic terminology pre-trusses in which all non-absorbing elements are regular are termed domains. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses and near-trusses as domains that satisfy the Ore condition are introduced and the pre-trusses of fractions are constructed through localisation. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces.Journal ArticlePublicacions Matemàtiques662683714Universitat Autonoma de Barcelona0214-14932014-43691720222022-07-0110.5565/publmat6622206COLLEGE NANMEMathematicsCOLLEGE CODESMASwansea UniversityThe research of Tomasz Brzezi´nski is partially supported by the National Science Centre, Poland, grant no. 2019/35/B/ST1/01115. The research of B. Rybo lowicz is supported by the EPSRC grant EP/V008129/1.2022-09-28T13:06:30.28586222021-02-03T18:27:29.7341932Faculty of Science and EngineeringSchool of Mathematics and Computer Science - MathematicsTomaszBrzezinski0000-0001-6270-34391StefanoMereta2BernardRybolowicz356171__19230__0e4be26c183c4566acc0c3fb33024161.pdfPrimie_revised.pdf2021-02-03T18:30:42.2959867Output382475application/pdfAccepted ManuscripttrueReleased with publisher's permission.trueeng 2022-09-28T13:06:30.2858622 v2 56171 2021-02-03 From pre-trusses to skew braces 30466d840b59627325596fbbb2c82754 0000-0001-6270-3439 Tomasz Brzezinski Tomasz Brzezinski true false 73f1af0073c82ade8fb20d1550e505d8 Stefano Mereta Stefano Mereta true false c821b8f926d58c3b7955b533a5c1472b Bernard Rybolowicz Bernard Rybolowicz true false 2021-02-03 SMA An algebraic system consisting of a set together with an associative binary and a ternary heap operations is studied. Such a system is termed a pre-truss and if a binary operation distributes over the heap operation on one side one speaks about a near-truss. If the binary operation in a near-truss is a group operation, then it can be specified or retracted to a skew brace, the notion introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534]. On the other hand if the binary operation in a near-truss has identity, then it gives rise to a skew-ring as introduced in [W. Rump, J. Algebra Appl. 18 (2019), 1950145]. Congruences in pre- and near-trusses are shown to arise from normal sub-heaps with an additional closure property of equivalence classes that involves both the ternary and binary operations. Such sub-heaps are called paragons. A necessary and sufficient criterion on paragonsunder which the quotient of a unital near-truss corresponds to a skew brace is derived. Regular elements in a pre-truss are defined as elements with left and right cancellation properties; following the ring-theoretic terminology pre-trusses in which all non-absorbing elements are regular are termed domains. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses and near-trusses as domains that satisfy the Ore condition are introduced and the pre-trusses of fractions are constructed through localisation. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces. Journal Article Publicacions Matemàtiques 66 2 683 714 Universitat Autonoma de Barcelona 0214-1493 2014-4369 1 7 2022 2022-07-01 10.5565/publmat6622206 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University The research of Tomasz Brzezi´nski is partially supported by the National Science Centre, Poland, grant no. 2019/35/B/ST1/01115. The research of B. Rybo lowicz is supported by the EPSRC grant EP/V008129/1. 2022-09-28T13:06:30.2858622 2021-02-03T18:27:29.7341932 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Tomasz Brzezinski 0000-0001-6270-3439 1 Stefano Mereta 2 Bernard Rybolowicz 3 56171__19230__0e4be26c183c4566acc0c3fb33024161.pdf Primie_revised.pdf 2021-02-03T18:30:42.2959867 Output 382475 application/pdf Accepted Manuscript true Released with publisher's permission. true eng From pre-trusses to skew braces From pre-trusses to skew braces Tomasz Brzezinski Stefano Mereta Bernard Rybolowicz From pre-trusses to skew braces From pre-trusses to skew braces From pre-trusses to skew braces From pre-trusses to skew braces From pre-trusses to skew braces 30466d840b59627325596fbbb2c82754 73f1af0073c82ade8fb20d1550e505d8 c821b8f926d58c3b7955b533a5c1472b 30466d840b59627325596fbbb2c82754_***_Tomasz Brzezinski 73f1af0073c82ade8fb20d1550e505d8_***_Stefano Mereta c821b8f926d58c3b7955b533a5c1472b_***_Bernard Rybolowicz Tomasz Brzezinski Stefano Mereta Bernard Rybolowicz Tomasz Brzezinski Stefano Mereta Bernard Rybolowicz Journal article Publicacions Matemàtiques 66 2 683 2022 Swansea University 0214-1493 2014-4369 10.5565/publmat6622206 Universitat Autonoma de Barcelona Faculty of Science and Engineering facultyofscienceandengineering Faculty of Science and Engineering facultyofscienceandengineering Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics 1 0 An algebraic system consisting of a set together with an associative binary and a ternary heap operations is studied. Such a system is termed a pre-truss and if a binary operation distributes over the heap operation on one side one speaks about a near-truss. If the binary operation in a near-truss is a group operation, then it can be specified or retracted to a skew brace, the notion introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534]. On the other hand if the binary operation in a near-truss has identity, then it gives rise to a skew-ring as introduced in [W. Rump, J. Algebra Appl. 18 (2019), 1950145]. Congruences in pre- and near-trusses are shown to arise from normal sub-heaps with an additional closure property of equivalence classes that involves both the ternary and binary operations. Such sub-heaps are called paragons. A necessary and sufficient criterion on paragonsunder which the quotient of a unital near-truss corresponds to a skew brace is derived. Regular elements in a pre-truss are defined as elements with left and right cancellation properties; following the ring-theoretic terminology pre-trusses in which all non-absorbing elements are regular are termed domains. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses and near-trusses as domains that satisfy the Ore condition are introduced and the pre-trusses of fractions are constructed through localisation. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces. 2022-07-01T04:10:56Z 1763753754402750464 10.969907