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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?

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1 Answer 1

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Hebisch and Weinert called such a structure a semifield in [1, p. 428, definition 1.5].

However, their terminology is based on a very loose definition of a semiring, which does not assume the existence of neutral elements and the commutativity of the addition. Nowadays, it looks like the following definition of a semiring is more or less accepted:

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 $r \in R$.

Moreover, according to Wikipedia, the term semifield has at least two conflicting meanings.

[1] Hebisch, Udo; Weinert, Hanns Joachim. Semirings and semifields. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996.

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