# Type or classification of a distributive algebraic structure made from a subsemigroup and a subgroup?

NB: $a,b,c \in S$

The following nontrivial (but trivial looking) algebraic structure $S$ is constructed. It was found that that $(S,+)$ is isomorphic to the left zero semigroup of 3 elements, while the $(S,\times)$ is isomorphic to $(\mathbb{Z}/3\mathbb{Z},+)$

Therefore $(S,\times)$ forms a subgroup with identity 1

Meanwhile, $(S,+)$ forms a subsemigroup where all elements in the structure are left absorbers/left zeros

Distributivity "trivially" holds due to all elements being left zeros, similarly for + associativity (All distributive and associative laws were checked independently for all 27 possible entries)

Attempt to classify it:

1. The structure is a ringoid as at least one distributive law holds
2. The structure is not a pseudoring as $(S,+)$ is not commutative

Is there a name in general for a structure with a set $A$ and two binary operators $\cdot,\circ$ such that $\circ$ distributes over $\cdot$, $(A,\circ)$ forms a subgroup while $(A,\cdot)$ forms a subsemigroup?

A semiring is a set $R$ equipped with two binary operations $+$ and $\cdot$ such that $(R, +)$ is a commutative monoid with identity element $0$, $(R, \cdot)$ is a monoid with identity element $1$, multiplication left and right distributes over addition and $0⋅r = r⋅0 = 0$ for all $r \in R$.