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From pre-trusses to skew braces
Publicacions Matemàtiques, Volume: 66, Issue: 2, Pages: 683 - 714
Swansea University Authors: Tomasz Brzezinski , Stefano Mereta, Bernard Rybolowicz
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DOI (Published version): 10.5565/publmat6622206
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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Universitat Autonoma de Barcelona
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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.
Faculty of Science and Engineering
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.