From pre-trusses to skew braces

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 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 paragons under 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.

Near braces and p$p$‐deformed braided groups

“…Example Let us consider the following near‐truss introduced in [32, p. 710]: $$$$\begin{equation*} \begin{aligned} \mathrm{Q}(O(i))= {\left\lbrace \frac{m}{2p+1}+ \frac{n}{2q+1}i\;|\; p,q\in {\mathbb {Z}}, \; \mbox{$m+n$ is an odd integer}\right\rbrace} \subset \mathbb {C}. \end{aligned} \end{equation*}$$$$Then $$\mathrm{Q}(O(i))$$ together with operations $$a+_ib=a-i+b$$ and $$a\,\circ\, b=a\cdot b$$ forms a near brace, where $$+,\cdot$$ are addition and multiplication of complex numbers, respectively.…”