Aspects of the category SKB of skew braces

Semidirect products of skew braces

“…The equivalence relations on a digroup (D, * , •) that are compatible with both group operations (that is, congruences of the algebra (D, * , •)) are in one-to-one correspondence with ideals I of D, that is, normal subgroups I of (D, * ) that are also normal subgroups of (D, •) and have the property that a * I = a • I for every a ∈ D (the cosets modulo I with respect to the two operations coincide) [7]. There is a forgetful functor U of the category DiGp of digroups into the category Set of sets, which assigns to each digroup D the set D. For an element a of R, let λ a : R → R denote the mapping defined by λ a (b) = ab for every b ∈ R. Then λ a is a mapping R → R, that is, a morphism in the category Set.…”

“…(8) ⇒ ( 7) If e is an idempotent endomorphism satisfying (8), then the image of e is B, and its corestriction e| B : D → B satisfies (7).…”