Normal congruence relations on division groupoids
Let be a semigroup. For such that , we say that is an associate of . A subgroup of which contains exactly one associate of each element of is called an associate subgroup of . It induces a unary operation in an obvious way, and we speak of a unary semigroup satisfying three simple axioms. A normal cryptogroup is a completely regular semigroup whose -relation is a congruence and is a normal band. Using the representation of as a strong semilattice of Rees matrix semigroups,...
In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all j ∈ I, the group G = Πi∈IGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.