While a ring is a set with two operations, (Abelian) group with respect to one and semigroup with respect to another with a standard two-sided distributivity law, there are also interesting examples of a “brace-like” distributivity law instead. Truss is a formalization/generalization of the usage of the alternative distributivity which uses a heap as a weakening of a concept of a group. Trusses appear in a number of subjects including the study of solutions to quantum Yang-Baxter equation, ideal ring extensions, Hopf-Galois extensions, inverse semigroups etc.
If we drop Abelianess of the “additive” group then we talk about a more general notion of a skew truss (which is therefore to near-ring roughly what a truss is to a ring).
A left truss is a heap $(A,[.,.,.])$ together with another “multiplicative” operation which distributes with the ternary operation of the heap from the left
A truss is a left and right truss.
A left truss is a left brace if it is Abelian group under the multiplicative operation.
Alternatively, we can use the binary operations only, together with some sort of a cocycle (measuring nonstandardness of distributive law). So let $(A,+)$ be a group (though additive, not necessarily commutative!) and $(A,\cdot)$ a semigroup. As usually, consider the associated heap operation of $(A,+)$,
(where $-$ means adding, in this order, the additive inverse). Then the following are equivalent:
$(A,[.,.,.])$ is a truss
$\exists \sigma: A\to A$ such that $a\cdot(b+c) = (a\cdot b)-\sigma(a)+(a\cdot c)$
$\exists\lambda : A\times A\to SA$ such that $a\cdot(b+c) = (a\cdot b) + \lambda(a,c)$
$\exists\mu : A\times A\to SA$ such that $a\cdot(b+c) = \mu(a,b)+(a\cdot c)$
$\exists\kappa,\tilde\kappa$ such that $a\cdot(b+c) = \kappa(a,b)+\tilde\kappa(a,c)$
Tomasz Brzeziński, Trusses: Between braces and rings, Trans. Amer. Math. Soc. 372 (2019), 4149-4176 doi; preprint pdf
T. Brzeziński, Enter truss, Northatlantic NCG seminar 2021 (video 2hr yt)
Tomasz Brzeziński, Bernard Rybołowicz, Modules over trusses vs modules over rings: direct sums and free modules, Algebra and Representation Theory 25, 1–23 (2022) doi
Ryszard R. Adruszkiewicz, Tomasz Brzeziński, Bernard Rybołowicz, Ideal ring extensions and trusses, arXiv:2101.09484
Last revised on August 22, 2022 at 18:27:41. See the history of this page for a list of all contributions to it.